138 Logical reasoning question on Data sufficiency

Here are some examples of Logical Reasoning questions on Data Sufficiency:

Example 1:

Question: Is the number xxx greater than 10?

Statement 1: xxx is a positive integer.

Statement 2: xxx is less than 20.

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 only tells us that xxx is positive but does not tell us if it’s greater than 10. So, this alone is not sufficient.
• Statement 2 tells us that xxx is less than 20, but doesn’t give any information about whether xxx is greater than 10. So, this alone is not sufficient.
• Combining both, we know that xxx is a positive integer less than 20, but this still doesn’t guarantee if xxx is greater than 10. Answer: E) Both statements together are not sufficient.

Example 2:

Question: What is the value of xxx?

Statement 1: 2x+3=9.

Statement 2: x=3.

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 provides an equation: 2x+3=9, solving it gives x=3. This alone is sufficient.
• Statement 2 directly gives x=3, which is also sufficient. Answer: D) Each statement alone is sufficient.

Example 3:

Question: Is y an even number?

Statement 1: y+5y is an odd number.

Statement 2: y is a prime number.

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 tells us that y+5 is odd. For y+5 to be odd, y must be even, because even + odd = odd.
• Statement 2 tells us that y is a prime number. The only even prime number is 2, so y must be 2, which is even. Thus, both statements provide sufficient information to determine that y is even. Answer: D) Each statement alone is sufficient.

Example 4:

Question: What is the perimeter of a rectangle?

Statement 1: The length of the rectangle is 12 units, and the width is 6 units.

Statement 2: The area of the rectangle is 72 square units.

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 directly gives the length and width, so the perimeter can be calculated as 2×(12+6)=36.
• Statement 2 gives the area, but doesn’t provide enough information to calculate the perimeter by itself. Answer: A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.

Example 5:

Question: Is the integer aaa divisible by 4?

Statement 1: aaa is divisible by 2.

Statement 2: aaa is divisible by 8.

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 only tells us that aaa is divisible by 2, which does not guarantee divisibility by 4.
• Statement 2 tells us that aaa is divisible by 8, and since 8 is a multiple of 4, aaa must be divisible by 4. Answer: B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.

Example 6:

Question: Is p+q=10?

Statement 1: p=4

Statement 2: q=6

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 gives p=4, but we don’t know q, so we cannot determine if p+q=10 .
• Statement 2 gives q=6, but we don’t know p, so we cannot determine if p+q=10.
• Combining both, p=4 and q=6, so p+q=10, which satisfies the equation. Answer: C) Both statements together are sufficient, but neither statement alone is sufficient.

Example 7:

Question: What is the value of z?

Statement 1: z2=49

Statement 2: z=−7

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 gives z2=49, so z=7 or z=−7, which means we cannot determine a unique value for z from this statement alone.
• Statement 2 gives z=−7, which is specific.
• Combining both, Statement 2 provides the exact value of zzz, making Statement 1 unnecessary. Answer: B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.

Example 8:

Question: Is x+y=15?

Statement 1: x=10

Statement 2: y=5

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 gives x=10, but we don’t know y, so we cannot determine if x+y=15.
• Statement 2 gives y=5, but we don’t know xxx, so we cannot determine if x+y=15.
• Combining both, x=10 and y=5, so x+y=15, which satisfies the equation. Answer: C) Both statements together are sufficient, but neither statement alone is sufficient.

Example 9:

Question: Is www an odd number?

Statement 1: w2 is an odd number.

Statement 2: w is a prime number.

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 tells us that w2 is odd. Since the square of an odd number is odd, www must be odd.
• Statement 2 tells us that www is a prime number, and the only even prime number is 2. Therefore, www must be an odd prime number.
• Since both statements independently indicate that www is odd, both are sufficient on their own. Answer: D) Each statement alone is sufficient.

Example 10:

Question: What is the value of mmm?

Statement 1: m+5=20

Statement 2: m−3=12

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 gives m+5=20, solving for mmm, we get m=15.
• Statement 2 gives m−3=12, solving for mmm, we get m=15.
• Both statements independently give us the value m=15.
• Answer: D) Each statement alone is sufficient.

Example 11:

Question: Is t>0?

Statement 1: t2=25

Statement 2: t is negative.

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 tells us that t2=25, so t=5 or t=−5. We cannot determine if t>0 from this alone.
• Statement 2 tells us that ttt is negative, so t=−5, which is not greater than 0.
• Thus, Statement 2 alone gives a definitive answer that ttt is negative and not greater than 0. Answer: B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.

Example 12:

Question: Is the integer nnn divisible by 6?

Statement 1: nnn is divisible by 2.

Statement 2: nnn is divisible by 3.

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 tells us that nnn is divisible by 2, but this does not guarantee that nnn is divisible by 6.
• Statement 2 tells us that nnn is divisible by 3, but this alone does not guarantee divisibility by 6.
• However, combining both, if nnn is divisible by both 2 and 3, it must be divisible by 6 (since 6 is the least common multiple of 2 and 3). Answer: C) Both statements together are sufficient, but neither statement alone is sufficient.

Example 13:

Question: What is the value of zzz?

Statement 1: 3z+5=20

Statement 2: z=5

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 gives the equation 3z+5=20, solving for z, we get z=5, which gives us the value of z.
• Statement 2 directly gives z=5, which is sufficient.
• Each statement alone gives us the same value for z, so both statements are sufficient individually. Answer: D) Each statement alone is sufficient.

Example 14:

Question: Is w>0?

Statement 1: w2>0

Statement 2: w≠0

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 tells us that w2>0, which means www could be either positive or negative, as squaring any non-zero number results in a positive value. This statement alone is not sufficient.
• Statement 2 tells us that w≠0, so www could be either positive or negative, but we still cannot definitively say if w>0. This statement alone is also not sufficient.
• Combining both statements, we know that w≠0 and w2>0, but still cannot determine whether w is positive or negative. Answer: E) Both statements together are not sufficient.

Example 15:

Question: Is xxx a prime number?

Statement 1: x>1

Statement 2: x is divisible by 2.

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 tells us that x>1x > 1x>1, but this doesn’t tell us if xxx is prime.
• Statement 2 tells us that xxx is divisible by 2. Since the only even prime number is 2, xxx could only be prime if x=2x = 2x=2. This statement alone is sufficient to conclude that x=2x = 2x=2, which is prime. Answer: B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.

Example 16:

Question: Is yyy a positive integer?

Statement 1: yyy is divisible by 3.

Statement 2: y2=9y^2 = 9y2=9

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 tells us that yyy is divisible by 3, but we don’t know whether yyy is positive or negative. This statement is not sufficient.
• Statement 2 tells us that y2=9, so y=3 or y=−3y = -3. Since we are asked whether y is positive, we know from this statement that y=3, which is a positive integer. Answer: B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.

Example 17:

Question: What is the perimeter of a square?

Statement 1: The area of the square is 64 square units.

Statement 2: The side length of the square is 8 units.

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 tells us that the area of the square is 64 square units. The area of a square is given by Area=s2\text, where s is the side length. So, s2=64, which gives s=8. The perimeter of a square is 4×s=4×8=324 \times s = 4 \times 8 = 324×s=4×8=32.
• Statement 2 directly gives the side length of the square as 8 units, so the perimeter is 4×8=324 \times 8 = 324×8=32. Answer: D) Each statement alone is sufficient.

Example 18:

Question: Is x greater than y?

Statement 1: x+y=10

Statement 2: x−y=4

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 tells us that x+y=10, but we cannot determine the relationship between xxx and y from this alone.
• Statement 2 tells us that x−y=4, but we still do not know the values of xxx and yyy and cannot determine if x>y from this alone.
• Combining both statements, we can solve the system of equations:
  1. x+y=10x−y=4
  Adding the two equations gives 2x=14, so x=7. Substituting into x+y=10, we get 7+y=107 , so y=3y . Therefore, x=7 and y=3, so x>y. Answer: C) Both statements together are sufficient, but neither statement alone is sufficient.

Example 19:

Question: Is mmm an integer?

Statement 1: m=3.5+2.5m

Statement 2: m=7

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 gives m=3.5+2.5=6m, which is an integer. So, Statement 1 alone is sufficient to determine that mmm is an integer.
• Statement 2 directly gives m=7, which is also an integer. Thus, Statement 2 alone is also sufficient to determine that mmm is an integer. Answer: D) Each statement alone is sufficient.

Example 20:

Question: Is nnn divisible by 4?

Statement 1: n=2×6n = 2 \times 6n=2×6

Statement 2: n=12n = 12n=12

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 gives n=2×6=12, which is divisible by 4.
• Statement 2 directly gives n=12, which is also divisible by 4. Answer: D) Each statement alone is sufficient.

Example 21:

Question: Is a>b?

Statement 1: a+b=10

Statement 2: a−b=4

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 tells us that a+b=10, but this alone doesn’t tell us the relative sizes of a and b.
• Statement 2 tells us that a−b=4, but by itself, it doesn’t give us enough information to compare a and b.
• Combining both statements:
  • a+b=10 and a−b=4.
  • Adding these two equations gives 2a=14, so a=7.
  • Substituting into a+b=10, we get 7+b=10, so b=3b.
  • Therefore, a=7 and b=3, and clearly a>b. Answer: C) Both statements together are sufficient, but neither statement alone is sufficient.

Example 22:

Question: What is the value of xxx?

Statement 1: x+6=12

Statement 2: x=6

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 gives x+6=12, solving for x, we get x=6x.
• Statement 2 directly gives x=6x.
• Both statements give the same value of x=6x. Answer: D) Each statement alone is sufficient.

Example 23:

Question: Is yyy divisible by 5?

Statement 1: y=15y

Statement 2: y is an odd number.

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 tells us that y=15, and since 15 is divisible by 5, we can conclude that y is divisible by 5.
• Statement 2 tells us that y is an odd number, but this doesn’t tell us whether y is divisible by 5.
• Statement 1 alone is sufficient. Answer: A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.

Example 24:

Question: Is mmm a positive integer?

Statement 1: m2=16

Statement 2: m is an integer.

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 tells us that m2=16, so m=4 or m=−4m. We cannot determine if mmm is positive from this alone.
• Statement 2 tells us that mmm is an integer, but this doesn’t tell us whether mmm is positive.
• Combining both statements, we still can’t definitively conclude if mmm is positive because mmm could be 4 or −4. Answer: E) Both statements together are not sufficient.

Example 25:

Question: What is the value of p?

Statement 1: p3=8

Statement 2: p is positive.

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 tells us that p3=8, so p=2, which gives the value of p.
• Statement 2 tells us that ppp is positive, but this alone doesn’t give us the exact value of ppp.
• Statement 1 alone is sufficient to determine that p=2. Answer: A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.

Example 26:

Question: Is the integer qqq divisible by both 3 and 4?

Statement 1: qqq is divisible by 12.

Statement 2: qqq is divisible by 6.

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 tells us that qqq is divisible by 12, which is divisible by both 3 and 4. Thus, qqq is divisible by both 3 and 4.
• Statement 2 tells us that qqq is divisible by 6, but this doesn’t guarantee divisibility by both 3 and 4 (for example, q=6q = 6q=6 is divisible by 3, but not by 4).
• Statement 1 alone is sufficient. Answer: A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.

Example 27:

Question: Is the integer rrr even?

Statement 1: rrr is divisible by 4.

Statement 2: rrr is divisible by 2.

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 tells us that rrr is divisible by 4, so rrr is even.
• Statement 2 tells us that rrr is divisible by 2, which means rrr is even as well.
• Both statements alone are sufficient to determine that rrr is even. Answer: D) Each statement alone is sufficient.

Example 28:

Question: Is zzz a multiple of 9?

Statement 1: zzz is divisible by 3.

Statement 2: z=27z = 27z=27

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 tells us that zzz is divisible by 3, but this does not guarantee that zzz is a multiple of 9.
• Statement 2 gives z=27z = 27z=27, which is a multiple of 9.
• Statement 2 alone is sufficient to answer the question. Answer: B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.

Example 29:

Question: What is the value of ttt?

Statement 1: t2=49t^2 = 49t2=49

Statement 2: ttt is a positive integer.

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 gives t2=49t^2 = 49t2=49, so t=7t = 7t=7 or t=−7t = -7t=−7.
• Statement 2 tells us that ttt is positive, so t=7t = 7t=7.
• Statement 2 alone is sufficient to determine the value of ttt. Answer: B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.

Example 30:

Question: What is the value of ppp?

Statement 1: p2=49p^2 = 49p2=49

Statement 2: ppp is negative.

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 gives p2=49p^2 = 49p2=49, so p=7p = 7p=7 or p=−7p = -7p=−7.
• Statement 2 tells us that ppp is negative, so p=−7p = -7p=−7.
• Combining both statements, we can determine that p=−7p = -7p=−7. Answer: C) Both statements together are sufficient, but neither statement alone is sufficient.

Example 31:

Question: Is kkk a multiple of 6?

Statement 1: kkk is divisible by 3.

Statement 2: kkk is divisible by 2.

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 tells us that kkk is divisible by 3, but this alone doesn’t guarantee that kkk is divisible by 6.
• Statement 2 tells us that kkk is divisible by 2, but this alone doesn’t guarantee that kkk is divisible by 6.
• Combining both statements: If kkk is divisible by both 2 and 3, then kkk must be divisible by 6. Answer: C) Both statements together are sufficient, but neither statement alone is sufficient.

Example 32:

Question: Is ttt a prime number?

Statement 1: ttt is divisible by 3.

Statement 2: t>1t > 1t>1

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 tells us that ttt is divisible by 3. This means that ttt could be 3, which is prime, or a multiple of 3, which is not prime.
• Statement 2 tells us that t>1t > 1t>1, but this doesn’t tell us whether ttt is prime.
• Combining both statements, we still can’t determine if ttt is prime because it could be any number greater than 1 divisible by 3 (e.g., t=6t = 6t=6). Answer: E) Both statements together are not sufficient.

Example 33:

Question: What is the value of nnn?

Statement 1: nnn is a positive integer.

Statement 2: n2=36n^2 = 36n2=36

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 tells us that nnn is a positive integer, but this alone doesn’t give us the value of nnn.
• Statement 2 gives us n2=36n^2 = 36n2=36, so n=6n = 6n=6 or n=−6n = -6n=−6. We cannot determine from this statement alone if nnn is positive or negative.
• Combining both statements, we know n2=36n^2 = 36n2=36 and nnn is positive, so n=6n = 6n=6. Answer: C) Both statements together are sufficient, but neither statement alone is sufficient.

Example 34:

Question: Is xxx an even integer?

Statement 1: xxx is divisible by 4.

Statement 2: xxx is divisible by 2.

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 tells us that xxx is divisible by 4. Any number divisible by 4 is even, so this statement is sufficient to conclude that xxx is even.
• Statement 2 tells us that xxx is divisible by 2, but this alone doesn’t confirm whether xxx is even or odd.
• Statement 1 alone is sufficient. Answer: A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.

Example 35:

Question: Is mmm a multiple of both 2 and 5?

Statement 1: mmm is divisible by 10.

Statement 2: mmm is divisible by 5.

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 tells us that mmm is divisible by 10. Since 10 is the product of both 2 and 5, this means that mmm is divisible by both 2 and 5, so this statement is sufficient.
• Statement 2 tells us that mmm is divisible by 5, but this alone doesn’t confirm that mmm is divisible by 2.
• Statement 1 alone is sufficient. Answer: A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.

Example 36:

Question: What is the value of zzz?

Statement 1: z+7=10z + 7 = 10z+7=10

Statement 2: z=3z = 3z=3

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 gives z+7=10z + 7 = 10z+7=10, solving for zzz, we get z=3z = 3z=3.
• Statement 2 directly gives z=3z = 3z=3.
• Both statements give the same value for zzz, so either statement alone is sufficient. Answer: D) Each statement alone is sufficient.

Example 37:

Question: Is ppp divisible by both 3 and 4?

Statement 1: ppp is divisible by 12.

Statement 2: ppp is divisible by 6.

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 tells us that ppp is divisible by 12. Since 12 is divisible by both 3 and 4, ppp must be divisible by both 3 and 4. Therefore, Statement 1 alone is sufficient.
• Statement 2 tells us that ppp is divisible by 6, but this doesn’t guarantee divisibility by 4.
• Statement 1 alone is sufficient. Answer: A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.

Example 38:

Question: Is qqq a prime number?

Statement 1: qqq is divisible by 2.

Statement 2: q=7q = 7q=7

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 tells us that qqq is divisible by 2, so q=2q = 2q=2, which is prime. Therefore, Statement 1 is sufficient to determine if qqq is prime.
• Statement 2 directly gives q=7q = 7q=7, which is a prime number.
• Both statements alone are sufficient to determine that qqq is prime. Answer: D) Each statement alone is sufficient.

Example 39:

Question: What is the value of xxx?

Statement 1: x2=16x^2 = 16×2=16

Statement 2: x>0x > 0x>0

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 gives x2=16x^2 = 16×2=16, so x=4x = 4x=4 or x=−4x = -4x=−4.
• Statement 2 tells us that x>0x > 0x>0, so x=4x = 4x=4.
• Combining both statements, we can determine that x=4x = 4x=4. Answer: C) Both statements together are sufficient, but neither statement alone is sufficient.

Example 40:

Question: Is yyy an even integer?

Statement 1: yyy is divisible by 2.

Statement 2: yyy is divisible by 4.

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 tells us that yyy is divisible by 2, so yyy is even, which directly answers the question.
• Statement 2 tells us that yyy is divisible by 4, and since any number divisible by 4 is also divisible by 2, this implies that yyy is even as well.
• Both statements alone are sufficient. Answer: D) Each statement alone is sufficient.

Example 41:

Question: What is the value of mmm?

Statement 1: m−5=10m – 5 = 10m−5=10

Statement 2: m=15m = 15m=15

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 gives m−5=10m – 5 = 10m−5=10, so m=15m = 15m=15.
• Statement 2 directly gives m=15m = 15m=15.
• Both statements give the same value for mmm, so either statement alone is sufficient. Answer: D) Each statement alone is sufficient.

Example 42:

Question: Is ppp divisible by 6?

Statement 1: ppp is divisible by 3.

Statement 2: ppp is divisible by 2.

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 tells us that ppp is divisible by 3, but this does not guarantee that ppp is divisible by 6.
• Statement 2 tells us that ppp is divisible by 2, but this does not guarantee that ppp is divisible by 6.
• Combining both statements, if ppp is divisible by both 2 and 3, then ppp must be divisible by 6. Answer: C) Both statements together are sufficient, but neither statement alone is sufficient.

Example 43:

Question: Is xxx an odd integer?

Statement 1: xxx is divisible by 3.

Statement 2: xxx is divisible by 2.

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 tells us that xxx is divisible by 3, but this alone does not tell us if xxx is odd or even.
• Statement 2 tells us that xxx is divisible by 2, which means xxx is even, so it cannot be odd.
• Statement 2 alone is sufficient to determine that xxx is not odd. Answer: B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.

Example 44:

Question: What is the value of zzz?

Statement 1: z+3=10z + 3 = 10z+3=10

Statement 2: zzz is an integer.

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 gives z+3=10z + 3 = 10z+3=10, solving for zzz, we get z=7z = 7z=7.
• Statement 2 tells us that zzz is an integer, but this doesn’t give us the value of zzz.
• Statement 1 alone is sufficient to determine the value of zzz. Answer: A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.

Example 45:

Question: Is a>ba > ba>b?

Statement 1: a+b=10a + b = 10a+b=10

Statement 2: a−b=4a – b = 4a−b=4

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 gives a+b=10a + b = 10a+b=10, but this alone does not give us a clear relationship between aaa and bbb.
• Statement 2 gives a−b=4a – b = 4a−b=4, but this alone does not help us determine the relationship between aaa and bbb either.
• Combining both statements:
  • From a+b=10a + b = 10a+b=10 and a−b=4a – b = 4a−b=4, adding these equations gives 2a=142a = 142a=14, so a=7a = 7a=7.
  • Substituting a=7a = 7a=7 into a+b=10a + b = 10a+b=10, we get 7+b=107 + b = 107+b=10, so b=3b = 3b=3.
  • Therefore, a=7a = 7a=7 and b=3b = 3b=3, and indeed a>ba > ba>b. Answer: C) Both statements together are sufficient, but neither statement alone is sufficient.

Example 46:

Question: Is xxx greater than 5?

Statement 1: x=10x = 10x=10

Statement 2: x>5x > 5x>5

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 directly gives x=10x = 10x=10, so xxx is indeed greater than 5.
• Statement 2 tells us that x>5x > 5x>5, which directly answers the question.
• Both statements alone are sufficient to conclude that xxx is greater than 5. Answer: D) Each statement alone is sufficient.

Example 47:

Question: Is zzz divisible by 15?

Statement 1: zzz is divisible by both 3 and 5.

Statement 2: z=45z = 45z=45

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 tells us that zzz is divisible by both 3 and 5. Since 15 is the product of 3 and 5, this means that zzz is divisible by 15.
• Statement 2 gives z=45z = 45z=45, and since 45 is divisible by 15, this statement alone is sufficient.
• Both statements are sufficient individually. Answer: D) Each statement alone is sufficient.

Example 48:

Question: What is the value of mmm?

Statement 1: m+4=20m + 4 = 20m+4=20

Statement 2: m=16m = 16m=16

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 gives m+4=20m + 4 = 20m+4=20, solving for mmm, we get m=16m = 16m=16.
• Statement 2 directly gives m=16m = 16m=16.
• Both statements give the same value for mmm, so either statement alone is sufficient. Answer: D) Each statement alone is sufficient.

Example 49:

Question: Is ttt an integer?

Statement 1: ttt is a rational number.

Statement 2: t=52t = \frac{5}{2}t=25​

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 tells us that ttt is a rational number, but a rational number can be either an integer or a fraction, so this statement does not confirm whether ttt is an integer.
• Statement 2 gives t=52t = \frac{5}{2}t=25​, which is a fraction, not an integer.
• Combining both statements, we still know that ttt is a rational number, but we also know that t=52t = \frac{5}{2}t=25​, which is not an integer. Answer: E) Both statements together are not sufficient.

Example 50:

Question: What is the value of aaa?

Statement 1: a2=25a^2 = 25a2=25

Statement 2: aaa is a negative number.

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 tells us that a2=25a^2 = 25a2=25, so a=5a = 5a=5 or a=−5a = -5a=−5.
• Statement 2 tells us that aaa is negative, so a=−5a = -5a=−5.
• Combining both statements, we can determine that a=−5a = -5a=−5. Answer: C) Both statements together are sufficient, but neither statement alone is sufficient.

Example 51:

Question: Is xxx a multiple of both 3 and 4?

Statement 1: xxx is divisible by 12.

Statement 2: xxx is divisible by 6.

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 tells us that xxx is divisible by 12. Since 12 is divisible by both 3 and 4, this means xxx is divisible by both 3 and 4. Therefore, Statement 1 alone is sufficient.
• Statement 2 tells us that xxx is divisible by 6, but this doesn’t necessarily mean that xxx is divisible by 4. Answer: A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.

Example 52:

Question: What is the value of yyy?

Statement 1: y=3x+2y = 3x + 2y=3x+2

Statement 2: x=5x = 5x=5

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 gives y=3x+2y = 3x + 2y=3x+2, but without knowing the value of xxx, we cannot determine yyy.
• Statement 2 tells us that x=5x = 5x=5. Substituting this into Statement 1, we get y=3(5)+2=15+2=17y = 3(5) + 2 = 15 + 2 = 17y=3(5)+2=15+2=17.
• Combining both statements, we can determine the value of yyy. Answer: C) Both statements together are sufficient, but neither statement alone is sufficient.

Example 53:

Question: Is mmm a positive integer?

Statement 1: mmm is divisible by 4.

Statement 2: mmm is divisible by 8.

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 tells us that mmm is divisible by 4. This does not tell us whether mmm is positive or negative, or whether mmm is an integer.
• Statement 2 tells us that mmm is divisible by 8. This also does not tell us whether mmm is positive or negative.
• Neither statement alone confirms if mmm is positive. Both statements together do not provide information about the sign of mmm. Answer: E) Both statements together are not sufficient.

Example 54:

Question: Is xxx greater than yyy?

Statement 1: x+y=10x + y = 10x+y=10

Statement 2: x−y=4x – y = 4x−y=4

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 gives x+y=10x + y = 10x+y=10, but this alone does not tell us if xxx is greater than yyy.
• Statement 2 gives x−y=4x – y = 4x−y=4, but this alone also does not tell us if xxx is greater than yyy.
• Combining both statements:
  • From x+y=10x + y = 10x+y=10 and x−y=4x – y = 4x−y=4, adding these two equations gives 2x=142x = 142x=14, so x=7x = 7x=7.
  • Substituting x=7x = 7x=7 into x+y=10x + y = 10x+y=10, we get 7+y=107 + y = 107+y=10, so y=3y = 3y=3.
  • Since x=7x = 7x=7 and y=3y = 3y=3, we can conclude that x>yx > yx>y. Answer: C) Both statements together are sufficient, but neither statement alone is sufficient.

Example 55:

Question: What is the value of ppp?

Statement 1: p3=125p^3 = 125p3=125

Statement 2: p=5p = 5p=5

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 gives p3=125p^3 = 125p3=125, so p=1253=5p = \sqrt[3]{125} = 5p=3125​=5.
• Statement 2 directly gives p=5p = 5p=5.
• Both statements give the same value for ppp, so either statement alone is sufficient. Answer: D) Each statement alone is sufficient.

Example 56:

Question: Is nnn a positive multiple of 7?

Statement 1: nnn is divisible by 14.

Statement 2: nnn is divisible by 7.

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 tells us that nnn is divisible by 14. Since 14 is a multiple of 7, nnn must also be divisible by 7. However, we don’t know if nnn is positive or negative.
• Statement 2 tells us that nnn is divisible by 7, but this does not tell us whether nnn is positive.
• Neither statement alone confirms whether nnn is positive. Answer: E) Both statements together are not sufficient.

Example 57:

Question: What is the value of xxx?

Statement 1: x+7=15x + 7 = 15x+7=15

Statement 2: xxx is a positive number.

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 gives x+7=15x + 7 = 15x+7=15, solving for xxx, we get x=8x = 8x=8.
• Statement 2 tells us that xxx is positive, but does not give us any additional information regarding its value.
• Statement 1 alone is sufficient to determine the value of xxx. Answer: A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.

Example 58:

Question: Is ppp an even integer?

Statement 1: p=2kp = 2kp=2k, where kkk is an integer.

Statement 2: p=3mp = 3mp=3m, where mmm is an integer.

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 tells us that p=2kp = 2kp=2k, where kkk is an integer. Since ppp is a multiple of 2, it is even.
• Statement 2 tells us that p=3mp = 3mp=3m, where mmm is an integer. This does not provide any direct information about whether ppp is even or odd.
• Statement 1 alone is sufficient to determine that ppp is even. Answer: A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.

Example 59:

Question: Is nnn divisible by 6?

Statement 1: nnn is divisible by both 2 and 3.

Statement 2: nnn is divisible by 12.

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 tells us that nnn is divisible by both 2 and 3. Since 6 is the product of 2 and 3, this means nnn is divisible by 6.
• Statement 2 tells us that nnn is divisible by 12, which means nnn is also divisible by 6, since 12 is a multiple of 6.
• Both statements are sufficient to determine that nnn is divisible by 6. Answer: D) Each statement alone is sufficient.

Example 60:

Question: What is the value of xxx?

Statement 1: x=3y+5x = 3y + 5x=3y+5

Statement 2: y=2y = 2y=2

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 gives x=3y+5x = 3y + 5x=3y+5, but we don’t know the value of yyy from Statement 1 alone.
• Statement 2 gives y=2y = 2y=2. Substituting this into Statement 1, we get x=3(2)+5=6+5=11x = 3(2) + 5 = 6 + 5 = 11x=3(2)+5=6+5=11.
• Combining both statements, we can determine the value of xxx. Answer: C) Both statements together are sufficient, but neither statement alone is sufficient.

Example 61:

Question: What is the value of zzz?

Statement 1: z2=36z^2 = 36z2=36

Statement 2: z=−6z = -6z=−6

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 gives z2=36z^2 = 36z2=36, so z=6z = 6z=6 or z=−6z = -6z=−6.
• Statement 2 directly gives z=−6z = -6z=−6, which answers the question.
• Statement 2 alone is sufficient to determine the value of zzz. Answer: B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.

Example 62:

Question: Is nnn divisible by 7?

Statement 1: n=49kn = 49kn=49k, where kkk is an integer.

Statement 2: n=7mn = 7mn=7m, where mmm is an integer.

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 tells us that n=49kn = 49kn=49k. Since 49 is divisible by 7, this means that nnn is divisible by 7.
• Statement 2 tells us that n=7mn = 7mn=7m, so nnn is divisible by 7 by definition.
• Both statements alone are sufficient to determine that nnn is divisible by 7. Answer: D) Each statement alone is sufficient.

Example 63:

Question: What is the value of ppp?

Statement 1: p2=64p^2 = 64p2=64

Statement 2: ppp is a negative integer.

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 gives p2=64p^2 = 64p2=64, so p=8p = 8p=8 or p=−8p = -8p=−8.
• Statement 2 tells us that ppp is a negative integer, so p=−8p = -8p=−8.
• Combining both statements, we can determine that p=−8p = -8p=−8. Answer: C) Both statements together are sufficient, but neither statement alone is sufficient.

Example 64:

Question: Is xxx an even integer?

Statement 1: x=2kx = 2kx=2k, where kkk is an integer.

Statement 2: x=4mx = 4mx=4m, where mmm is an integer.

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 tells us that x=2kx = 2kx=2k, where kkk is an integer. Since xxx is a multiple of 2, it is even.
• Statement 2 tells us that x=4mx = 4mx=4m, where mmm is an integer. Since 4 is a multiple of 2, xxx is even as well.
• Both statements alone confirm that xxx is even. Answer: D) Each statement alone is sufficient.

Example 65:

Question: Is xxx greater than 5?

Statement 1: x=3y+2x = 3y + 2x=3y+2

Statement 2: y=3y = 3y=3

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 gives x=3y+2x = 3y + 2x=3y+2, but we don’t know the value of yyy from Statement 1 alone.
• Statement 2 gives y=3y = 3y=3. Substituting this into Statement 1, we get x=3(3)+2=9+2=11x = 3(3) + 2 = 9 + 2 = 11x=3(3)+2=9+2=11.
• Combining both statements, we can determine that x=11x = 11x=11, which is greater than 5. Answer: C) Both statements together are sufficient, but neither statement alone is sufficient.

Example 66:

Question: Is xxx a prime number?

Statement 1: x=2y+1x = 2y + 1x=2y+1, where yyy is an integer.

Statement 2: xxx is greater than 5.

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 tells us that x=2y+1x = 2y + 1x=2y+1, which means xxx is always an odd number, but we don’t know whether xxx is prime or not (since odd numbers can be either prime or composite).
• Statement 2 tells us that xxx is greater than 5, but this doesn’t tell us whether xxx is prime or not.
• Combining both statements doesn’t guarantee that xxx is prime, as xxx could still be a composite odd number greater than 5 (like 9, 15, etc.). Answer: E) Both statements together are not sufficient.

Example 67:

Question: Is yyy divisible by 5?

Statement 1: y=25ky = 25ky=25k, where kkk is an integer.

Statement 2: yyy is divisible by 10.

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 tells us that y=25ky = 25ky=25k, which means yyy is divisible by 5, since 25 is a multiple of 5.
• Statement 2 tells us that yyy is divisible by 10, and since 10 is a multiple of 5, yyy must also be divisible by 5.
• Both statements alone are sufficient to conclude that yyy is divisible by 5. Answer: D) Each statement alone is sufficient.

Example 68:

Question: What is the value of mmm?

Statement 1: m+4=10m + 4 = 10m+4=10

Statement 2: m=6m = 6m=6

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 gives m+4=10m + 4 = 10m+4=10, solving for mmm, we get m=6m = 6m=6.
• Statement 2 directly gives m=6m = 6m=6.
• Both statements confirm that m=6m = 6m=6, so each statement alone is sufficient. Answer: D) Each statement alone is sufficient.

Example 69:

Question: Is ppp an odd number?

Statement 1: p=2k+1p = 2k + 1p=2k+1, where kkk is an integer.

Statement 2: ppp is greater than 1.

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 tells us that p=2k+1p = 2k + 1p=2k+1, which is the general form of an odd number. Therefore, ppp is definitely odd.
• Statement 2 only tells us that ppp is greater than 1, but this doesn’t confirm whether ppp is odd or even.
• Statement 1 alone is sufficient to determine that ppp is odd. Answer: A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.

Example 70:

Question: What is the value of xxx?

Statement 1: x=2y+3x = 2y + 3x=2y+3

Statement 2: y=4y = 4y=4

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 gives x=2y+3x = 2y + 3x=2y+3, but we need to know the value of yyy to determine xxx.
• Statement 2 gives y=4y = 4y=4. Substituting this into Statement 1, we get x=2(4)+3=8+3=11x = 2(4) + 3 = 8 + 3 = 11x=2(4)+3=8+3=11.
• Combining both statements, we can determine that x=11x = 11x=11. Answer: C) Both statements together are sufficient, but neither statement alone is sufficient.

Example 71:

Question: Is nnn a perfect square?

Statement 1: n=64n = 64n=64

Statement 2: nnn is divisible by 4.

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 gives n=64n = 64n=64, which is a perfect square (828^282).
• Statement 2 tells us that nnn is divisible by 4, but this does not tell us if nnn is a perfect square. nnn could be 4, 16, or any other number divisible by 4.
• Statement 1 alone is sufficient to determine that nnn is a perfect square. Answer: A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.

Example 72:

Question: Is xxx divisible by both 3 and 4?

Statement 1: xxx is divisible by 12.

Statement 2: xxx is divisible by 6.

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 tells us that xxx is divisible by 12. Since 12 is divisible by both 3 and 4, xxx is also divisible by both 3 and 4.
• Statement 2 tells us that xxx is divisible by 6, but this does not guarantee divisibility by 4, as xxx could be divisible by 6 but not by 4.
• Statement 1 alone is sufficient to confirm divisibility by both 3 and 4. Answer: A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.

Example 73:

Question: Is nnn a multiple of 9?

Statement 1: nnn is divisible by 18.

Statement 2: nnn is divisible by 3.

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 tells us that nnn is divisible by 18, and since 18 is a multiple of 9, nnn is divisible by 9.
• Statement 2 tells us that nnn is divisible by 3, but this does not necessarily imply divisibility by 9.
• Statement 1 alone is sufficient to confirm that nnn is a multiple of 9. Answer: A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.

Example 74:

Question: What is the value of xxx?

Statement 1: x2=36x^2 = 36×2=36

Statement 2: xxx is negative.

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 tells us that x2=36x^2 = 36×2=36, which implies that x=6x = 6x=6 or x=−6x = -6x=−6.
• Statement 2 tells us that xxx is negative, so from Statement 1, we can conclude that x=−6x = -6x=−6.
• Combining both statements, we can determine that x=−6x = -6x=−6. Answer: C) Both statements together are sufficient, but neither statement alone is sufficient.

Example 75:

Question: Is mmm divisible by both 2 and 3?

Statement 1: mmm is divisible by 6.

Statement 2: mmm is divisible by 3.

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 tells us that mmm is divisible by 6. Since 6 is divisible by both 2 and 3, mmm must be divisible by both 2 and 3.
• Statement 2 tells us that mmm is divisible by 3, but it doesn’t guarantee divisibility by 2.
• Statement 1 alone is sufficient to conclude that mmm is divisible by both 2 and 3. Answer: A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.

Example 76:

Question: What is the value of ppp?

Statement 1: p2=49p^2 = 49p2=49

Statement 2: ppp is an even integer.

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 tells us that p2=49p^2 = 49p2=49, so p=7p = 7p=7 or p=−7p = -7p=−7.
• Statement 2 tells us that ppp is an even integer, but neither 7 nor -7 is even.
• Combining both statements leads to a contradiction, as the first statement gives us two possible values for ppp, neither of which is even. Answer: E) Both statements together are not sufficient.

Example 77:

Question: Is nnn a multiple of 8?

Statement 1: nnn is divisible by 16.

Statement 2: nnn is divisible by 4.

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 tells us that nnn is divisible by 16. Since 16 is a multiple of 8, nnn must also be divisible by 8.
• Statement 2 tells us that nnn is divisible by 4, but divisibility by 4 alone doesn’t guarantee divisibility by 8.
• Statement 1 alone is sufficient to confirm that nnn is divisible by 8. Answer: A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.

Example 78:

Question: What is the value of yyy?

Statement 1: y=2x+4y = 2x + 4y=2x+4

Statement 2: x=6x = 6x=6

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 gives y=2x+4y = 2x + 4y=2x+4, but we don’t know the value of xxx from Statement 1 alone.
• Statement 2 tells us that x=6x = 6x=6. Substituting into Statement 1, we get y=2(6)+4=12+4=16y = 2(6) + 4 = 12 + 4 = 16y=2(6)+4=12+4=16.
• Combining both statements, we can determine that y=16y = 16y=16. Answer: C) Both statements together are sufficient, but neither statement alone is sufficient.

Example 79:

Question: Is xxx divisible by both 2 and 3?

Statement 1: x=6kx = 6kx=6k, where kkk is an integer.

Statement 2: xxx is divisible by 2.

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 tells us that x=6kx = 6kx=6k, which means xxx is divisible by both 2 and 3 since 6 is divisible by both 2 and 3.
• Statement 2 tells us that xxx is divisible by 2, but this alone does not guarantee divisibility by 3.
• Statement 1 alone is sufficient to conclude that xxx is divisible by both 2 and 3. Answer: A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.

Example 80:

Question: What is the value of zzz?

Statement 1: z=3y+4z = 3y + 4z=3y+4

Statement 2: y=2y = 2y=2

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 gives z=3y+4z = 3y + 4z=3y+4, but we need to know the value of yyy to determine zzz.
• Statement 2 gives y=2y = 2y=2. Substituting into Statement 1, we get z=3(2)+4=6+4=10z = 3(2) + 4 = 6 + 4 = 10z=3(2)+4=6+4=10.
• Combining both statements, we can determine that z=10z = 10z=10. Answer: C) Both statements together are sufficient, but neither statement alone is sufficient.

Example 81:

Question: Is xxx divisible by 4?

Statement 1: x=12kx = 12kx=12k, where kkk is an integer.

Statement 2: x=16mx = 16mx=16m, where mmm is an integer.

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 tells us that x=12kx = 12kx=12k. Since 12 is divisible by 4, xxx must also be divisible by 4.
• Statement 2 tells us that x=16mx = 16mx=16m. Since 16 is divisible by 4, xxx must also be divisible by 4.
• Both statements alone confirm that xxx is divisible by 4. Answer: D) Each statement alone is sufficient.

Example 82:

Question: What is the value of ppp?

Statement 1: p+3=10p + 3 = 10p+3=10

Statement 2: p=7p = 7p=7

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 gives p+3=10p + 3 = 10p+3=10, which simplifies to p=7p = 7p=7.
• Statement 2 directly gives p=7p = 7p=7.
• Both statements confirm that p=7p = 7p=7, so each statement alone is sufficient. Answer: D) Each statement alone is sufficient.

Example 83:

Question: Is xxx divisible by both 5 and 6?

Statement 1: xxx is divisible by 30.

Statement 2: xxx is divisible by 5.

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 tells us that xxx is divisible by 30. Since 30 is divisible by both 5 and 6, xxx must also be divisible by both 5 and 6.
• Statement 2 tells us that xxx is divisible by 5, but it does not guarantee divisibility by 6.
• Statement 1 alone is sufficient to conclude that xxx is divisible by both 5 and 6. Answer: A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.

Example 84:

Question: Is mmm an even number?

Statement 1: m=2km = 2km=2k, where kkk is an integer.

Statement 2: mmm is divisible by 4.

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 gives m=2km = 2km=2k, which means mmm is even (since it’s a multiple of 2).
• Statement 2 tells us that mmm is divisible by 4, which also means mmm is even, but it does not provide new information about the nature of mmm’s evenness.
• Statement 1 alone is sufficient to conclude that mmm is even. Answer: A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.

Example 85:

Question: What is the value of xxx?

Statement 1: x−3=7x – 3 = 7x−3=7

Statement 2: x=10x = 10x=10

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 gives x−3=7x – 3 = 7x−3=7, which simplifies to x=10x = 10x=10.
• Statement 2 directly gives x=10x = 10x=10.
• Both statements confirm that x=10x = 10x=10, so each statement alone is sufficient. Answer: D) Each statement alone is sufficient.

Example 86:

Question: Is yyy a multiple of 4?

Statement 1: yyy is divisible by 8.

Statement 2: yyy is divisible by 2.

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 tells us that yyy is divisible by 8. Since 8 is a multiple of 4, yyy must also be divisible by 4.
• Statement 2 tells us that yyy is divisible by 2, but this does not guarantee divisibility by 4.
• Statement 1 alone is sufficient to conclude that yyy is divisible by 4. Answer: A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.

Example 87:

Question: Is nnn greater than 5?

Statement 1: n+2=7n + 2 = 7n+2=7

Statement 2: n=6n = 6n=6

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 gives n+2=7n + 2 = 7n+2=7, which simplifies to n=5n = 5n=5. Therefore, nnn is not greater than 5.
• Statement 2 directly gives n=6n = 6n=6, which is greater than 5.
• Statement 2 alone is sufficient to conclude that nnn is greater than 5. Answer: B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.

Example 88:

Question: What is the value of zzz?

Statement 1: z=3y+5z = 3y + 5z=3y+5

Statement 2: y=2y = 2y=2

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 gives z=3y+5z = 3y + 5z=3y+5, but we need to know the value of yyy to determine zzz.
• Statement 2 gives y=2y = 2y=2. Substituting into Statement 1, we get z=3(2)+5=6+5=11z = 3(2) + 5 = 6 + 5 = 11z=3(2)+5=6+5=11.
• Combining both statements, we can determine that z=11z = 11z=11. Answer: C) Both statements together are sufficient, but neither statement alone is sufficient.

Example 89:

Question: Is xxx an even number?

Statement 1: x=2kx = 2kx=2k, where kkk is an integer.

Statement 2: x=3mx = 3mx=3m, where mmm is an integer.

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 tells us that x=2kx = 2kx=2k, which means xxx is even.
• Statement 2 tells us that x=3mx = 3mx=3m, which only implies that xxx is divisible by 3 but not necessarily even.
• Statement 1 alone is sufficient to conclude that xxx is even. Answer: A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.

Example 90:

Question: What is the value of ttt?

Statement 1: t=5k+2t = 5k + 2t=5k+2, where kkk is an integer.

Statement 2: t=12t = 12t=12

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 gives t=5k+2t = 5k + 2t=5k+2, which describes ttt in terms of kkk, but we cannot determine ttt without knowing the value of kkk.
• Statement 2 directly gives t=12t = 12t=12.
• Statement 2 alone is sufficient to determine that t=12t = 12t=12. Answer: B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.

Example 91:

Question: Is xxx an integer?

Statement 1: x=3.5×2x = 3.5 \times 2x=3.5×2

Statement 2: x=7x = 7x=7

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 gives x=3.5×2=7x = 3.5 \times 2 = 7x=3.5×2=7, so xxx is an integer.
• Statement 2 directly gives x=7x = 7x=7, which is an integer.
• Both statements confirm that x=7x = 7x=7, which is an integer, so each statement alone is sufficient. Answer: D) Each statement alone is sufficient.

Example 92:

Question: What is the value of xxx?

Statement 1: 2x+3=112x + 3 = 112x+3=11

Statement 2: x=4x = 4x=4

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 gives the equation 2x+3=112x + 3 = 112x+3=11. Solving for xxx, we get 2x=82x = 82x=8, so x=4x = 4x=4.
• Statement 2 directly gives x=4x = 4x=4.
• Both statements confirm that x=4x = 4x=4, so each statement alone is sufficient. Answer: D) Each statement alone is sufficient.

Example 93:

Question: Is xxx divisible by both 3 and 5?

Statement 1: xxx is divisible by 15.

Statement 2: xxx is divisible by 3.

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 tells us that xxx is divisible by 15. Since 15 is divisible by both 3 and 5, xxx must be divisible by both 3 and 5.
• Statement 2 tells us that xxx is divisible by 3, but this alone doesn’t guarantee divisibility by 5.
• Statement 1 alone is sufficient to conclude that xxx is divisible by both 3 and 5. Answer: A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.

Example 94:

Question: Is yyy greater than 10?

Statement 1: y=2x+5y = 2x + 5y=2x+5

Statement 2: x=4x = 4x=4

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 gives y=2x+5y = 2x + 5y=2x+5, but we don’t know the value of xxx from Statement 1 alone.
• Statement 2 tells us that x=4x = 4x=4. Substituting this into Statement 1, we get y=2(4)+5=8+5=13y = 2(4) + 5 = 8 + 5 = 13y=2(4)+5=8+5=13, so y=13y = 13y=13, which is greater than 10.
• Combining both statements, we can conclude that y=13y = 13y=13, which is greater than 10. Answer: C) Both statements together are sufficient, but neither statement alone is sufficient.

Example 95:

Question: What is the value of mmm?

Statement 1: m+2=10m + 2 = 10m+2=10

Statement 2: m=8m = 8m=8

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 gives m+2=10m + 2 = 10m+2=10, which simplifies to m=8m = 8m=8.
• Statement 2 directly gives m=8m = 8m=8.
• Both statements confirm that m=8m = 8m=8, so each statement alone is sufficient. Answer: D) Each statement alone is sufficient.

Example 96:

Question: Is zzz divisible by 6?

Statement 1: z=3kz = 3kz=3k, where kkk is an integer.

Statement 2: z=2mz = 2mz=2m, where mmm is an integer.

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 tells us that z=3kz = 3kz=3k, meaning zzz is divisible by 3.
• Statement 2 tells us that z=2mz = 2mz=2m, meaning zzz is divisible by 2.
• Combining both statements, we see that zzz is divisible by both 2 and 3, so it is divisible by 6.
• Both statements together are sufficient, but neither statement alone is sufficient. Answer: C) Both statements together are sufficient, but neither statement alone is sufficient.

Example 97:

Question: What is the value of aaa?

Statement 1: a+b=10a + b = 10a+b=10

Statement 2: b=5b = 5b=5

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 gives a+b=10a + b = 10a+b=10, but we don’t know the value of bbb.
• Statement 2 tells us that b=5b = 5b=5. Substituting this into Statement 1, we get a+5=10a + 5 = 10a+5=10, so a=5a = 5a=5.
• Combining both statements, we can conclude that a=5a = 5a=5. Answer: C) Both statements together are sufficient, but neither statement alone is sufficient.

Example 98:

Question: Is xxx a prime number?

Statement 1: x=7x = 7x=7

Statement 2: x=2+5x = 2 + 5x=2+5

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 gives x=7x = 7x=7, which is a prime number.
• Statement 2 gives x=2+5=7x = 2 + 5 = 7x=2+5=7, which is also a prime number.
• Both statements confirm that x=7x = 7x=7, which is prime, so each statement alone is sufficient. Answer: D) Each statement alone is sufficient.

Example 99:

Question: Is mmm divisible by 9?

Statement 1: m=3nm = 3nm=3n, where nnn is an integer.

Statement 2: m=9pm = 9pm=9p, where ppp is an integer.

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 tells us that m=3nm = 3nm=3n, so mmm is divisible by 3, but it does not guarantee divisibility by 9.
• Statement 2 tells us that m=9pm = 9pm=9p, so mmm is divisible by 9.
• Statement 2 alone is sufficient to conclude that mmm is divisible by 9. Answer: B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.

Example 100:

Question: What is the value of ppp?

Statement 1: 2p=122p = 122p=12

Statement 2: p=6p = 6p=6

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 gives 2p=122p = 122p=12, which simplifies to p=6p = 6p=6.
• Statement 2 directly gives p=6p = 6p=6.
• Both statements confirm that p=6p = 6p=6, so each statement alone is sufficient. Answer: D) Each statement alone is sufficient.

Example 101:

Question: Is xxx greater than 10?

Statement 1: x=3y+4x = 3y + 4x=3y+4

Statement 2: y=5y = 5y=5

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 gives x=3y+4x = 3y + 4x=3y+4, but we do not know the value of yyy.
• Statement 2 gives y=5y = 5y=5. Substituting this into Statement 1, we get x=3(5)+4=15+4=19x = 3(5) + 4 = 15 + 4 = 19x=3(5)+4=15+4=19, which is greater than 10.
• Combining both statements, we can conclude that x=19x = 19x=19, which is greater than 10. Answer: C) Both statements together are sufficient, but neither statement alone is sufficient.

Example 102:

Question: Is ppp divisible by 12?

Statement 1: p=6qp = 6qp=6q, where qqq is an integer.

Statement 2: p=12rp = 12rp=12r, where rrr is an integer.

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 tells us that p=6qp = 6qp=6q, meaning ppp is divisible by 6, but it does not guarantee divisibility by 12.
• Statement 2 tells us that p=12rp = 12rp=12r, meaning ppp is divisible by 12.
• Statement 2 alone is sufficient to conclude that ppp is divisible by 12. Answer: B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.

Example 103:

Question: What is the value of yyy?

Statement 1: 2y+3=92y + 3 = 92y+3=9

Statement 2: y=3y = 3y=3

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 gives the equation 2y+3=92y + 3 = 92y+3=9. Solving for yyy, we get 2y=62y = 62y=6, so y=3y = 3y=3.
• Statement 2 directly gives y=3y = 3y=3.
• Both statements confirm that y=3y = 3y=3, so each statement alone is sufficient. Answer: D) Each statement alone is sufficient.

Example 104:

Question: Is xxx a multiple of 4?

Statement 1: x=2kx = 2kx=2k, where kkk is an integer.

Statement 2: x=4mx = 4mx=4m, where mmm is an integer.

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 tells us that x=2kx = 2kx=2k, so xxx is divisible by 2, but it does not guarantee that xxx is divisible by 4.
• Statement 2 tells us that x=4mx = 4mx=4m, so xxx is divisible by 4.
• Statement 2 alone is sufficient to conclude that xxx is a multiple of 4. Answer: B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.

Example 105:

Question: What is the value of ttt?

Statement 1: t=4x+3t = 4x + 3t=4x+3

Statement 2: x=2x = 2x=2

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 gives t=4x+3t = 4x + 3t=4x+3, but we need to know the value of xxx to determine ttt.
• Statement 2 tells us that x=2x = 2x=2. Substituting this into Statement 1, we get t=4(2)+3=8+3=11t = 4(2) + 3 = 8 + 3 = 11t=4(2)+3=8+3=11.
• Combining both statements, we can determine that t=11t = 11t=11. Answer: C) Both statements together are sufficient, but neither statement alone is sufficient.

Example 106:

Question: Is aaa greater than bbb?

Statement 1: a=b+5a = b + 5a=b+5

Statement 2: b=3b = 3b=3

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 gives a=b+5a = b + 5a=b+5, which means aaa is always 5 greater than bbb, so aaa is greater than bbb.
• Statement 2 tells us that b=3b = 3b=3, but it doesn’t provide enough information on its own to compare aaa and bbb.
• Statement 1 alone is sufficient to conclude that aaa is greater than bbb. Answer: A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.

Example 107:

Question: Is xxx divisible by 10?

Statement 1: x=5yx = 5yx=5y, where yyy is an integer.

Statement 2: x=2kx = 2kx=2k, where kkk is an integer.

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 tells us that x=5yx = 5yx=5y, meaning xxx is divisible by 5, but it does not guarantee divisibility by 10.
• Statement 2 tells us that x=2kx = 2kx=2k, meaning xxx is divisible by 2, but it does not guarantee divisibility by 10.
• Combining both statements, we can conclude that xxx is divisible by both 2 and 5, so it is divisible by 10.
• Both statements together are sufficient, but neither statement alone is sufficient. Answer: C) Both statements together are sufficient, but neither statement alone is sufficient.

Example 108:

Question: Is yyy an even number?

Statement 1: y=2ky = 2ky=2k, where kkk is an integer.

Statement 2: y=3my = 3my=3m, where mmm is an integer.

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 gives y=2ky = 2ky=2k, so yyy is even.
• Statement 2 gives y=3my = 3my=3m, but this only tells us that yyy is divisible by 3, not necessarily even.
• Statement 1 alone is sufficient to conclude that yyy is even. Answer: A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.

Example 109:

Question: Is xxx a positive integer?

Statement 1: x=3yx = 3yx=3y, where yyy is an integer.

Statement 2: x=−6x = -6x=−6

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 tells us that x=3yx = 3yx=3y, but it doesn’t specify whether xxx is positive or negative.
• Statement 2 tells us that x=−6x = -6x=−6, which is not a positive integer.
• Statement 2 alone is sufficient to conclude that xxx is not a positive integer. Answer: B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.

Example 110:

Question: Is ppp an odd number?

Statement 1: p=2k+1p = 2k + 1p=2k+1, where kkk is an integer.

Statement 2: p=5p = 5p=5

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 gives p=2k+1p = 2k + 1p=2k+1, which is the form of an odd number.
• Statement 2 tells us that p=5p = 5p=5, which is an odd number.
• Both statements confirm that ppp is an odd number, so each statement alone

Example 111:

Question: Is nnn a prime number?

Statement 1: nnn is an odd number.

Statement 2: nnn is greater than 2.

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 tells us that nnn is odd, but odd numbers can be either prime (e.g., 3, 5, 7) or non-prime (e.g., 9, 15).
• Statement 2 tells us that nnn is greater than 2, but there are many numbers greater than 2 that are not prime (e.g., 4, 6, 8).
• Combining both statements, we still cannot determine whether nnn is prime because it could be a non-prime odd number greater than 2 (e.g., 9, 15). Answer: E) Both statements together are not sufficient.

Example 112:

Question: What is the value of xxx?

Statement 1: x+4=10x + 4 = 10x+4=10

Statement 2: x=6x = 6x=6

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 gives the equation x+4=10x + 4 = 10x+4=10, which simplifies to x=6x = 6x=6.
• Statement 2 directly gives x=6x = 6x=6.
• Both statements confirm that x=6x = 6x=6, so each statement alone is sufficient. Answer: D) Each statement alone is sufficient.

Example 113:

Question: Is yyy divisible by 6?

Statement 1: y=2ky = 2ky=2k, where kkk is an integer.

Statement 2: y=3my = 3my=3m, where mmm is an integer.

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 tells us that yyy is divisible by 2.
• Statement 2 tells us that yyy is divisible by 3.
• Combining both statements, we can conclude that yyy is divisible by both 2 and 3, so yyy is divisible by 6. Answer: C) Both statements together are sufficient, but neither statement alone is sufficient.

Example 114:

Question: Is xxx greater than 100?

Statement 1: x=50+60x = 50 + 60x=50+60

Statement 2: x=90+25x = 90 + 25x=90+25

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 gives x=50+60=110x = 50 + 60 = 110x=50+60=110, which is greater than 100.
• Statement 2 gives x=90+25=115x = 90 + 25 = 115x=90+25=115, which is also greater than 100.
• Both statements confirm that xxx is greater than 100, so each statement alone is sufficient. Answer: D) Each statement alone is sufficient.

Example 115:

Question: Is mmm a negative number?

Statement 1: m=−5m = -5m=−5

Statement 2: m=5m = 5m=5

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 directly gives m=−5m = -5m=−5, which is a negative number.
• Statement 2 directly gives m=5m = 5m=5, which is a positive number.
• Statement 1 alone is sufficient to conclude that mmm is negative. Answer: A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.

Example 116:

Question: Is ppp divisible by 3?

Statement 1: p=9kp = 9kp=9k, where kkk is an integer.

Statement 2: p=6mp = 6mp=6m, where mmm is an integer.

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 tells us that p=9kp = 9kp=9k, so ppp is divisible by 9, which is divisible by 3.
• Statement 2 tells us that p=6mp = 6mp=6m, so ppp is divisible by 6, which is also divisible by 3.
• Both statements confirm that ppp is divisible by 3, so each statement alone is sufficient. Answer: D) Each statement alone is sufficient.

Example 117:

Question: What is the value of ttt?

Statement 1: t+7=12t + 7 = 12t+7=12

Statement 2: t=5t = 5t=5

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 gives t+7=12t + 7 = 12t+7=12, which simplifies to t=5t = 5t=5.
• Statement 2 directly gives t=5t = 5t=5.
• Both statements confirm that t=5t = 5t=5, so each statement alone is sufficient. Answer: D) Each statement alone is sufficient.

Example 118:

Question: Is mmm a perfect square?

Statement 1: m=25m = 25m=25

Statement 2: m=36m = 36m=36

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 gives m=25m = 25m=25, which is a perfect square (since 52=255^2 = 2552=25).
• Statement 2 gives m=36m = 36m=36, which is also a perfect square (since 62=366^2 = 3662=36).
• Both statements confirm that mmm is a perfect square, so each statement alone is sufficient. Answer: D) Each statement alone is sufficient.

Example 119:

Question: Is ppp an integer?

Statement 1: p=102p = \frac{10}{2}p=210​

Statement 2: p=72p = \frac{7}{2}p=27​

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 gives p=102=5p = \frac{10}{2} = 5p=210​=5, which is an integer.
• Statement 2 gives p=72=3.5p = \frac{7}{2} = 3.5p=27​=3.5, which is not an integer.
• Statement 1 alone is sufficient to conclude that ppp is an integer, while Statement 2 is not. Answer: A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.

Example 120:

Question: Is xxx a positive number?

Statement 1: x=−4x = -4x=−4

Statement 2: x=5x = 5x=5

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 gives x=−4x = -4x=−4, which is not a positive number.
• Statement 2 gives x=5x = 5x=5, which is a positive number.
• Statement 2 alone is sufficient to conclude that xxx is positive. Answer: B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.

Example 121:

Question: Is xxx an even number?

Statement 1: x=2yx = 2yx=2y, where yyy is an integer.

Statement 2: x=3yx = 3yx=3y, where yyy is an integer.

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 tells us that x=2yx = 2yx=2y, which is the form of an even number.
• Statement 2 tells us that x=3yx = 3yx=3y, but this does not guarantee that xxx is even because yyy can be either odd or even.
• Statement 1 alone is sufficient to conclude that xxx is even. Answer: A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.

Example 122:

Question: Is ppp greater than 100?

Statement 1: p=50+60p = 50 + 60p=50+60

Statement 2: p=90+25p = 90 + 25p=90+25

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 gives p=50+60=110p = 50 + 60 = 110p=50+60=110, which is greater than 100.
• Statement 2 gives p=90+25=115p = 90 + 25 = 115p=90+25=115, which is also greater than 100.
• Both statements confirm that ppp is greater than 100, so each statement alone is sufficient. Answer: D) Each statement alone is sufficient.

Example 123:

Question: Is nnn divisible by 6?

Statement 1: n=2kn = 2kn=2k, where kkk is an integer.

Statement 2: n=3mn = 3mn=3m, where mmm is an integer.

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 tells us that nnn is divisible by 2.
• Statement 2 tells us that nnn is divisible by 3.
• Combining both statements, we can conclude that nnn is divisible by both 2 and 3, so nnn is divisible by 6. Answer: C) Both statements together are sufficient, but neither statement alone is sufficient.

Example 124:

Question: Is xxx greater than 50?

Statement 1: x=30+25x = 30 + 25x=30+25

Statement 2: x=45+10x = 45 + 10x=45+10

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 gives x=30+25=55x = 30 + 25 = 55x=30+25=55, which is greater than 50.
• Statement 2 gives x=45+10=55x = 45 + 10 = 55x=45+10=55, which is also greater than 50.
• Both statements confirm that xxx is greater than 50, so each statement alone is sufficient. Answer: D) Each statement alone is sufficient.

Example 125:

Question: Is xxx divisible by 4?

Statement 1: x=8kx = 8kx=8k, where kkk is an integer.

Statement 2: x=4mx = 4mx=4m, where mmm is an integer.

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 tells us that x=8kx = 8kx=8k, which is divisible by 4.
• Statement 2 tells us that x=4mx = 4mx=4m, which is also divisible by 4.
• Both statements confirm that xxx is divisible by 4, so each statement alone is sufficient. Answer: D) Each statement alone is sufficient.

Example 126:

Question: Is xxx greater than 5?

Statement 1: x=3+4x = 3 + 4x=3+4

Statement 2: x=2+2x = 2 + 2x=2+2

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 gives x=3+4=7x = 3 + 4 = 7x=3+4=7, which is greater than 5.
• Statement 2 gives x=2+2=4x = 2 + 2 = 4x=2+2=4, which is not greater than 5.
• Statement 1 alone is sufficient to conclude that xxx is greater than 5. Answer: A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.

Example 127:

Question: Is yyy a positive integer?

Statement 1: y=2ky = 2ky=2k, where kkk is an integer.

Statement 2: y=−3y = -3y=−3

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 tells us that y=2ky = 2ky=2k, so yyy is an integer, but it does not guarantee that yyy is positive.
• Statement 2 tells us that y=−3y = -3y=−3, which is not a positive integer.
• Statement 2 alone is sufficient to conclude that yyy is not a positive integer. Answer: B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.

Example 128:

Question: Is mmm divisible by 2?

Statement 1: m=4km = 4km=4k, where kkk is an integer.

Statement 2: m=5km = 5km=5k, where kkk is an integer.

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 tells us that m=4km = 4km=4k, so mmm is divisible by 2 because 4 is divisible by 2.
• Statement 2 tells us that m=5km = 5km=5k, but this does not guarantee divisibility by 2 because 5 is not divisible by 2.
• Statement 1 alone is sufficient to conclude that mmm is divisible by 2. Answer: A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.

Example 129:

Question: Is nnn greater than 0?

Statement 1: n=2kn = 2kn=2k, where kkk is an integer.

Statement 2: n=−1n = -1n=−1

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 tells us that n=2kn = 2kn=2k, which means nnn could be any integer multiple of 2, including negative, zero, or positive numbers.
• Statement 2 tells us that n=−1n = -1n=−1, which is less than 0.
• Neither statement alone can confirm whether nnn is greater than 0. Answer: E) Both statements together are not sufficient.

Example 130:

Question: Is ppp divisible by 3?

Statement 1: p=9kp = 9kp=9k, where kkk is an integer.

Statement 2: p=6mp = 6mp=6m, where mmm is an integer.

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 tells us that p=9kp = 9kp=9k, which is divisible by 3 because 9 is divisible by 3.
• Statement 2 tells us that p=6mp = 6mp=6m, which is also divisible by 3 because 6 is divisible by 3.
• Both statements confirm that ppp is divisible by 3, so each statement alone is sufficient. Answer: D) Each statement alone is sufficient.

Example 131:

Question: Is yyy a prime number?

Statement 1: yyy is a multiple of 5.

Statement 2: y>5y > 5y>5.

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 tells us that yyy is a multiple of 5, so yyy could be 5, 10, 15, etc. However, only 5 is prime, so this alone does not confirm that yyy is prime.
• Statement 2 tells us that y>5y > 5y>5, but it doesn’t give any information about whether yyy is prime.
• Combining both statements, we still cannot determine if yyy is prime (for example, y=10y = 10y=10 is not prime but is greater than 5). Answer: E) Both statements together are not sufficient.

Example 132:

Question: Is xxx greater than 100?

Statement 1: x=50+70x = 50 + 70x=50+70

Statement 2: x=100+1x = 100 + 1x=100+1

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 gives x=50+70=120x = 50 + 70 = 120x=50+70=120, which is greater than 100.
• Statement 2 gives x=100+1=101x = 100 + 1 = 101x=100+1=101, which is also greater than 100.
• Both statements confirm that xxx is greater than 100, so each statement alone is sufficient. Answer: D) Each statement alone is sufficient.

Example 133:

Question: Is zzz a positive integer?

Statement 1: z=−3z = -3z=−3

Statement 2: z=3z = 3z=3

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 tells us that z=−3z = -3z=−3, which is not a positive integer.
• Statement 2 tells us that z=3z = 3z=3, which is a positive integer.
• Statement 2 alone is sufficient to conclude that zzz is positive. Answer: B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.

Example 134:

Question: Is xxx divisible by 12?

Statement 1: x=36kx = 36kx=36k, where kkk is an integer.

Statement 2: x=24mx = 24mx=24m, where mmm is an integer.

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 tells us that x=36kx = 36kx=36k, so xxx is divisible by 36, which is divisible by 12.
• Statement 2 tells us that x=24mx = 24mx=24m, so xxx is divisible by 24, which is also divisible by 12.
• Both statements confirm that xxx is divisible by 12, so each statement alone is sufficient. Answer: D) Each statement alone is sufficient.

Example 135:

Question: Is ppp greater than 50?

Statement 1: p=2×30p = 2 \times 30p=2×30

Statement 2: p=60p = 60p=60

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 gives p=2×30=60p = 2 \times 30 = 60p=2×30=60, which is greater than 50.
• Statement 2 gives p=60p = 60p=60, which is also greater than 50.
• Both statements confirm that ppp is greater than 50, so each statement alone is sufficient. Answer: D) Each statement alone is sufficient.

Example 136:

Question: Is nnn divisible by 3?

Statement 1: n=9kn = 9kn=9k, where kkk is an integer.

Statement 2: n=6mn = 6mn=6m, where mmm is an integer.

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 tells us that n=9kn = 9kn=9k, which is divisible by 3 because 9 is divisible by 3.
• Statement 2 tells us that n=6mn = 6mn=6m, which is also divisible by 3 because 6 is divisible by 3.
• Both statements confirm that nnn is divisible by 3, so each statement alone is sufficient. Answer: D) Each statement alone is sufficient.

Example 137:

Question: Is mmm greater than 0?

Statement 1: m=−10m = -10m=−10

Statement 2: m=2+3m = 2 + 3m=2+3

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 tells us that m=−10m = -10m=−10, which is not greater than 0.
• Statement 2 gives m=2+3=5m = 2 + 3 = 5m=2+3=5, which is greater than 0.
• Statement 2 alone is sufficient to conclude that mmm is greater than 0. Answer: B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.

Example 138:

Question: Is xxx divisible by both 2 and 3?

Statement 1: x=6kx = 6kx=6k, where kkk is an integer.

Statement 2: x=12mx = 12mx=12m, where mmm is an integer.

Answer Choices:

• A) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• B) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• C) Both statements together are sufficient, but neither statement alone is sufficient.
• D) Each statement alone is sufficient.
• E) Both statements together are not sufficient.

Solution:

• Statement 1 tells us that x=6kx = 6kx=6k, so xxx is divisible by both 2 and 3 because 6 is divisible by both.
• Statement 2 tells us that x=12mx = 12mx=12m, so xxx is divisible by both 2 and 3 because 12 is divisible by both.
• Both statements confirm that xxx is divisible by both 2 and 3, so each statement alone is sufficient. Answer: D) Each statement alone is sufficient.

Also read : Logical verification of truth of statement

Please join discussion on Facebook about world facts and its secret.