GMAT Quiz: Determine Sufficiency

What this quiz covers

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

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
4. 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 2

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (correct answer)
2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
4. 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 3

1. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
2. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (correct answer)
3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
4. 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 $x$ or $y$ 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 $x$ or $y$ 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 4

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (correct answer)
4. 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 5

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (correct answer)
2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
4. 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.