Determine Sufficiency - GMAT Data Insights

1. If $x$ is nonzero, is $x$ positive? (1) $x^2 > 1$ (2) $2^x > 2$
   Answer: Only (2) sufficient 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.
2. If n is an integer, is n divisible by 6? (1) $n^2$ is divisible by 6 (2) $3n$ is divisible by 18
   Answer: Each statement alone suffices 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.
3. Tickets cost $5 or $8. How many $8 tickets were sold? (1) Total revenue was $156. (2) 24 tickets were sold.
   Answer: Both, neither alone 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.
4. 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
   Answer: BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. 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.
5. 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 $A = 60°$
   Answer: EACH statement ALONE is sufficient to answer the question. 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.
6. If $a$ and $b$ are positive integers, is $a/b > 1$? (1) $a > 2b$ (2) $a^2 > b$
   Answer: Only (1) sufficient 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.
7. If $x$ and $y$ are integers, is $xy$ even? (1) $x + y$ is odd (2) $x^2 + y^2$ is even
   Answer: Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. 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.
8. Is the average of $a$, $b$, and $c$ equal to $b$? (1) $a + c = 2b$ (2) $b$ is the median of $a$, $b$, and $c$
   Answer: Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. 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.
9. If $x$ and $y$ are integers, are both $x$ and $y$ even? (1) $x + y$ is even (2) $xy$ is even
   Answer: Together, but neither alone 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.
10. If $x$ is a real number, is $x < 0$? (1) $x^3 < x^2$ (2) $x^2 < x$
    Answer: Statement (2) only suffices 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.
11. 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.
    Answer: Both, neither alone 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.
12. Is positive integer $n$ divisible by 12? (1) $3n$ is divisible by 36. (2) $n^2$ is divisible by 144.
    Answer: Each statement alone 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.
13. 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
    Answer: Statement (2) only suffices 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.
14. What is the value of $x^2 + y^2$? (1) $x + y = 5$ and $xy = 6$ (2) $(x - y)^2 = 1$
    Answer: Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. 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.

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