GMAT Quiz: Determine Sufficiency

14 questions · exam conditions

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Determine SufficiencyQuestion 1 of 14

If k is a positive integer, what is the remainder when $7^k$ is divided by 5? (1) k is even (2) k is a multiple of 4

Statement (1) only suffices

Statement (2) only suffices

Together, but neither alone

Each statement alone suffices

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GMAT Quiz

GMAT Quiz: Determine Sufficiency

Practice Determine Sufficiency in GMAT with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

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This quiz focuses on Determine Sufficiency, giving you a quick way to practice the rules, question types, and explanations that matter most for GMAT.

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Question 1

If k is a positive integer, what is the remainder when $7^k$ is divided by 5? (1) k is even (2) k is a multiple of 4

Statement (1) only suffices
Statement (2) only suffices (correct answer)
Together, but neither alone
Each statement alone suffices

Explanation: 7 mod 5 is 2, so the remainder of 7^k is the same as 2^k mod 5. Powers of 2 cycle through 2, 4, 3, 1 as k advances, so the remainder depends on k mod 4. Statement (2) makes k mod 4 = 0, giving remainder 1, so it suffices. Statement (1) is not enough: even k can be 2 or 4, giving remainders 4 or 1, respectively.

Question 2

If n is an integer, is n divisible by 6? (1) $n^2$ is divisible by 6 (2) $3n$ is divisible by 18

Statement (1) only suffices
Statement (2) only suffices
Together, but neither alone
Each statement alone suffices (correct answer)

Explanation: Because 6 = 2 times 3, a number is divisible by 6 exactly when it is divisible by both 2 and 3. If n^2 is divisible by 6, then 2 and 3 each divide n^2, so each must divide n; thus n is divisible by 6. For statement (2), 3n divisible by 18 means 3n = 18k, so n = 6k, again sufficient. The tempting wrong answer is 'together, but neither alone,' but each statement alone already forces n to be a multiple of 6.

Question 3

If $x$ and $y$ are integers, are both $x$ and $y$ even? (1) $x + y$ is even (2) $xy$ is even

Statement (1) only suffices
Statement (2) only suffices
Together, but neither alone (correct answer)
Each statement alone suffices

Explanation: x + y even means x and y have the same parity: both even or both odd. xy even means at least one is even. Alone, either fact allows mixed possibilities, so neither suffices. Together, same parity plus at least one even forces both even. The tempting mistake is thinking xy even alone proves both are even, but it only proves one is even.

Question 4

If $x$ is a real number, is $x < 0$ ? (1) $x^3 < x^2$ (2) $x^2 < x$

Statement (2) only suffices (correct answer)
Statement (1) only suffices
Together, but neither alone
Each statement alone suffices

Explanation: From (1), x^3 < x^2 gives x^2(x-1)<0, so x<1 with x not 0. That includes negatives but also numbers like 0.5, so it doesn't prove x<0. From (2), x^2 < x gives x(x-1)<0, so 0<x<1; x is positive, so the answer is no. Thus statement (2) alone suffices.

Question 5

Is positive integer $n$ divisible by 12? (1) $3n$ is divisible by 36. (2) $n^2$ is divisible by 144.

Only (1) sufficient
Only (2) sufficient
Both, neither alone
Each statement alone (correct answer)

Explanation: From (1), 3n divisible by 36 means 3n / 36 = n / 12 is an integer, so n is divisible by 12. From (2), 144 = 12^2; if n^2 contains all factors of 144, then n must contain at least 2^2 and 3, so n is divisible by 12. The tempting wrong view is that n^2 divisible by a square only suggests n might not be; but because n is an integer, those factors must be present in n. Each statement alone is sufficient.

Question 6

At constant rates, how long does pipe A take alone? (1) Together A and B fill the tank in 4 hr. (2) B alone takes 9 hr.

Only (1) sufficient
Only (2) sufficient
Both, neither alone (correct answer)
Each statement alone

Explanation: Let A's time be a and B's time be b. Statement (1) gives 1/a + 1/b = 1/4 but has two unknowns, so A alone isn't determined. Statement (2) gives b = 9, but without A's rate it tells you nothing about A. Together, 1/a + 1/9 = 1/4, so 1/a = 5/36 and a = 36/5 = 7.2 hours. The tempting error is taking the 4-hour combined time as A's time alone; you need B's rate to isolate A.

Question 7

If $x$ is nonzero, is $x$ positive? (1) $x^2 > 1$ (2) $2^x > 2$

Only (1) sufficient
Only (2) sufficient (correct answer)
Both, neither alone
Each statement alone

Explanation: Statement (1) gives x > 1 or x < -1, so x could be negative or positive; it is not sufficient. Statement (2) means 2^x > 2^1, so x > 1 because the base-2 exponential is increasing. That makes x positive, so only statement (2) is sufficient. The tempting error is treating x^2 > 1 as if it meant x > 1, forgetting negative values less than -1.

Question 8

Tickets cost $5 or $8. How many $8 tickets were sold? (1) Total revenue was $156. (2) 24 tickets were sold.

Only (1) sufficient
Only (2) sufficient
Both, neither alone (correct answer)
Each statement alone

Explanation: Let x be $5 tickets and y be $8 tickets. Revenue alone gives 5x + 8y = 156, with several possible whole-number pairs (for example, 20 and 7 or 4 and 17), so it is not enough. Ticket count alone gives x + y = 24 but no price mix. Together, substitute x = 24 - y: 5(24 - y) + 8y = 156, so 120 + 3y = 156 and y = 12. The tempting wrong choice is revenue alone, because it seems price-based but allows multiple ticket combinations.

Question 9

If $a$ and $b$ are positive integers, is $a/b > 1$ ? (1) $a > 2b$ (2) $a^2 > b$

Only (1) sufficient (correct answer)
Only (2) sufficient
Both, neither alone
Each statement alone

Explanation: Since b is positive, a/b > 1 exactly means a > b. Statement (1) gives a > 2b, and because b > 0, 2b > b, so a > b; sufficient. Statement (2), a^2 > b, doesn't force a > b: for example a = 3 and b = 8 gives 9 > 8 but 3/8 < 1. So only statement (1) is sufficient.

Question 10

In triangle $ABC$ , what is the length of side $BC$ ?

(1) The area of triangle $ABC$ is 24, and the altitude from $A$ to side $BC$ has length 6 (2) $AB = 10$ , $AC = 8$ , and angle $A = 60°$

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
EACH statement ALONE is sufficient to answer the question. (correct answer)

Explanation: Statement (1): Using the formula for area,

\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}

. Here,

24 = \frac{1}{2} \times BC \times 6

, which gives us

BC = 8

. Statement (1) is sufficient. Statement (2): We can use the Law of Cosines:

BC^2 = AB^2 + AC^2 - 2(AB)(AC)\cos A = 10^2 + 8^2 - 2(10)(8)\cos(60°) = 100 + 64 - 160 \times \frac{1}{2} = 164 - 80 = 84

. Therefore,

BC = \sqrt{84} = 2\sqrt{21}

. Statement (2) is sufficient. Each statement alone is sufficient.

Question 11

What is the value of $x^2 + y^2$ ?

(1) $x + y = 5$ and $xy = 6$ (2) $(x - y)^2 = 1$

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (correct answer)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
EACH statement ALONE is sufficient to answer the question.

Explanation: We need to find

x^2 + y^2

. Statement (1): Given

x + y = 5

and

xy = 6

. We can use the identity

(x + y)^2 = x^2 + 2xy + y^2

, which gives us

25 = x^2 + 12 + y^2

. Therefore,

x^2 + y^2 = 13

. Statement (1) is sufficient. Statement (2):

(x - y)^2 = 1

means

x^2 - 2xy + y^2 = 1

, so

x^2 + y^2 = 1 + 2xy

. However, we don't know the value of

xy

, so we cannot determine

x^2 + y^2

. Statement (2) alone is not sufficient.

Question 12

If $x$ and $y$ are integers, is $xy$ even?

(1) $x + y$ is odd (2) $x^2 + y^2$ is even

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (correct answer)
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
EACH statement ALONE is sufficient to answer the question.

Explanation: This data sufficiency question tests your understanding of even/odd properties and how they interact through different operations. The key insight is recognizing what each statement tells you about the parity (even/odd nature) of

x

and

y

. For

xy

to be even, at least one of

x

y

must be even. If both are odd, their product is odd. Statement (1) tells us

x + y

is odd. This happens only when one number is even and the other is odd (since even + even = even, and odd + odd = even). If exactly one of

x

y

is even, then their product

xy

must be even. Statement (1) alone is sufficient. Statement (2) tells us

x^2 + y^2

is even. Since squaring preserves parity (even² = even, odd² = odd), this means we have either even² + even² = even or odd² + odd² = even. Both scenarios give an even sum, so we can't determine whether

x

and

y

have the same parity or different parities. For example, if

x = 1, y = 3

, then

x^2 + y^2 = 10

(even) but

xy = 3

(odd). If

x = 1, y = 2

, then

x^2 + y^2 = 5

(odd), contradicting our statement. Actually, if

x = 2, y = 4

, then

x^2 + y^2 = 20

(even) and

xy = 8

(even). Statement (2) alone is insufficient. Answer choice B is correct because statement (1) alone is sufficient while statement (2) alone is not. Choices A and D incorrectly assess statement (2) as sufficient. Choice C incorrectly claims both statements are needed. Strategy tip: For even/odd questions, systematically test what each parity combination produces, and remember that addition and multiplication have different parity rules.

Question 13

What is the remainder when the positive integer $n$ is divided by 8?

(1) When $n$ is divided by 4, the remainder is 3 (2) When $n$ is divided by 6, the remainder is 5

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (correct answer)
EACH statement ALONE is sufficient to answer the question.

Explanation: Statement (1): If

n \equiv 3 \pmod{4}

, then

n = 4k + 3

for some integer

k

. When we divide by 8, we need to consider

k

even or odd. If

k = 2m

, then

n = 8m + 3

, so remainder is 3. If

k = 2m + 1

, then

n = 8m + 7

, so remainder is 7. Statement (1) alone is not sufficient. Statement (2): If

n \equiv 5 \pmod{6}

, then

n = 6j + 5

for some integer

j

. We need to find

n \bmod 8

. Since

\gcd(6,8) = 2

, the possible remainders when

n = 6j + 5

is divided by 8 depend on

j \bmod 4

. Testing:

j = 0: n = 5

, remainder = 5.

j = 1: n = 11

, remainder = 3.

j = 2: n = 17

, remainder = 1.

j = 3: n = 23

, remainder = 7. Statement (2) alone is not sufficient. Combined: We need

n \equiv 3 \pmod{4}

and

n \equiv 5 \pmod{6}

. Using the Chinese Remainder Theorem, since

\gcd(4,6) = 2

and

3 \equiv 5 \equiv 1 \pmod{2}

(the system is consistent), there exists a unique solution modulo

\text{lcm}(4,6) = 12

. Solving:

n = 4k + 3 = 6j + 5

. From the first equation,

k = \frac{n-3}{4}

. Substituting into

n \equiv 5 \pmod{6}

: we need

4k + 3 \equiv 5 \pmod{6}

, so

4k \equiv 2 \pmod{6}

, which gives

2k \equiv 1 \pmod{3}

. Since

2 \times 2 \equiv 1 \pmod{3}

, we have

k \equiv 2 \pmod{3}

. So

k = 3m + 2

, giving

n = 4(3m + 2) + 3 = 12m + 11

. Therefore,

n \equiv 11 \equiv 3 \pmod{8}

. Both statements together are sufficient.

Question 14

Is the average of $a$ , $b$ , and $c$ equal to $b$ ?

(1) $a + c = 2b$ (2) $b$ is the median of $a$ , $b$ , and $c$

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (correct answer)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
EACH statement ALONE is sufficient to answer the question.

Explanation: We want to determine if

\frac{a + b + c}{3} = b

, which is equivalent to asking if

a + b + c = 3b

, or

a + c = 2b

. Statement (1): This directly gives us

a + c = 2b

, which means

a + b + c = 3b

, so the average equals

b

. Statement (1) is sufficient. Statement (2): If

b

is the median of

a

b

, and

c

, then when the three numbers are arranged in order,

b

is in the middle. However, this doesn't guarantee that

a + c = 2b

. For example, if

a = 1

b = 2

c = 10

, then

b

is the median, but

a + c = 11 \neq 4 = 2b

. The average would be

\frac{13}{3} \neq 2

. Statement (2) alone is not sufficient.