Hard GRE Data Sufficiency Questions Practice Questions

Concept Explanation

GRE Data Sufficiency questions evaluate your ability to determine whether the provided information is sufficient to answer a specific mathematical question without requiring you to calculate a final numerical result.

While the standard GRE General Test primarily uses Quantitative Comparison, many advanced math evaluations and the GMAT-style Data Sufficiency format (often found in GRE Prep materials) require a high level of logical rigor. In these problems, you are given a question followed by two statements. You must decide if Statement 1 alone is sufficient, Statement 2 alone is sufficient, both together are sufficient, each independently is sufficient, or neither is sufficient even when combined. Success with Hard GRE Data Sufficiency Questions relies on identifying "number properties" traps, such as considering negative numbers, fractions, and zero, rather than just positive integers.

To solve these efficiently, follow a consistent process: analyze the question stem for constraints, evaluate Statement 1, evaluate Statement 2 independently, and only combine them if neither is sufficient on its own. This systematic approach prevents the common mistake of "carrying over" information from one statement into the evaluation of the other.

Solved Examples

1. Is the integer $n$ divisible by 12?

   Statement 1: $n$ is divisible by 3.

   Statement 2: $n$ is divisible by 4.

   Solution:

   1. Analyze Statement 1: If $n = 3$, the answer is No. If $n = 12$, the answer is Yes. Insufficient.
   2. Analyze Statement 2: If $n = 4$, the answer is No. If $n = 12$, the answer is Yes. Insufficient.
   3. Combine: If $n$ is divisible by both 3 and 4, it must be divisible by their least common multiple, which is 12. Therefore, $n$ must be divisible by 12.
   4. Final Answer: Both statements together are sufficient.
2. What is the value of $x + y$?

   Statement 1: $x^2 - y^2 = 24$

   Statement 2: $x - y = 4$

   Solution:

   1. Analyze Statement 1: $x^2 - y^2$ factors into $(x-y)(x+y) = 24$. Without knowing either factor, we cannot find $x+y$. Insufficient.
   2. Analyze Statement 2: Knowing $x - y = 4$ tells us nothing about the sum $x + y$. Insufficient.
   3. Combine: Substitute Statement 2 into the factored form of Statement 1: $(4)(x+y) = 24$. Dividing by 4 gives $x+y = 6$.
   4. Final Answer: Both statements together are sufficient.
3. If $x$ and $y$ are non-zero integers, is $x^y$ a positive integer?

   Statement 1: $x$ is positive.

   Statement 2: $y$ is an even integer.

   Solution:

   1. Analyze Statement 1: If $x = 2$ and $y = -1$, $x^y = \frac{1}{2}$ (not an integer). If $x = 2$ and $y = 2$, $x^y = 4$ (is a positive integer). Insufficient.
   2. Analyze Statement 2: If $y = 2$ and $x = -2$, $x^y = 4$. If $y = -2$ and $x = -2$, $x^y = \frac{1}{4}$. Insufficient.
   3. Combine: If $x$ is positive and $y$ is even, $x^y$ could still be a fraction if $y$ is a negative even integer (e.g., $2^{-2} = \frac{1}{4}$).
   4. Final Answer: Statements 1 and 2 together are still insufficient.

Practice Questions

1. If $a$ and $b$ are integers, is $a + b$ an odd integer?

Statement 1: $3a + 4b$ is even.

Statement 2: $5a + 2b$ is odd.

2. What is the value of the real number $z$?

Statement 1: $z^2 - 6z + 9 = 0$

Statement 2: $z > 0$

3. Is the average (arithmetic mean) of $m, n, p, \text{ and } q$ equal to 15?

Statement 1: $m + n + p = 45$

Statement 2: $q = 15$

4. Is $x > y$?

Statement 1: $x^2 > y^2$

Statement 2: $x - y > 0$

5. A certain jar contains only red marbles and blue marbles. What is the probability of drawing a red marble?

Statement 1: The ratio of red marbles to blue marbles is 3:2.

Statement 2: There are 20 more red marbles than blue marbles.

6. Is the integer $k$ a prime number?

Statement 1: $9 < k < 15$

Statement 2: $(k - 1)! + 1$ is divisible by $k$.

7. What is the value of $x$?

Statement 1: $|x - 5| = 2$

Statement 2: $|x| = 7$

8. If $x$ is a positive integer, is $\sqrt{x}$ an integer?

Statement 1: $4x$ is a perfect square.

Statement 2: $3x$ is not a perfect square.

9. Does the line $L$ pass through the origin (0,0)?

Statement 1: The line $L$ passes through the points (3, 4) and (-3, -4).

Statement 2: The equation of line $L$ is $y = mx$.

10. Is $xy < 0$?

Statement 1: $x^3 y^2 < 0$

Statement 2: $x + y > 0$

Answers & Explanations

1. Answer: Statement 2 alone is sufficient.
   For Statement 1, if $3a + 4b$ is even, $3a$ must be even (since $4b$ is always even), so $a$ is even. $b$ can be odd or even. For Statement 2, if $5a + 2b$ is odd, $5a$ must be odd (since $2b$ is even), so $a$ is odd. If $a$ is odd, $a+b$ depends on $b$. Wait, let's re-evaluate: In Statement 2, $5a$ is odd, so $a$ is odd. But we still don't know $b$. Re-checking Statement 1: $a$ is even, $b$ unknown. If we combine them? No, Statement 2 actually doesn't give $b$. Let's look at the standard parity rules. Actually, in Statement 2, $2b$ is always even, so $5a$ must be odd for the sum to be odd. This means $a$ is odd. This doesn't reveal $b$. Both statements together? $a$ is even (from S1) and $a$ is odd (from S2) is a contradiction, meaning these statements cannot describe the same $a$. In DS, statements never contradict. Let's re-read: $3a$ even means $a$ even. $5a$ odd means $a$ odd. If the question was "is $a$ odd?", S2 would be sufficient. For $a+b$, we need both. This question is a classic parity trap.
2. Answer: Statement 1 alone is sufficient.
   Statement 1 factors to $(z-3)^2 = 0$, which means $z = 3$. This is a single value. Statement 2 says $z > 0$, which allows infinitely many values.
3. Answer: Both statements together are sufficient.
   The average is $\frac{m+n+p+q}{4}$. Statement 1 gives $m+n+p = 45$. Statement 2 gives $q = 15$. Together, $\frac{45+15}{4} = \frac{60}{4} = 15$.
4. Answer: Statement 2 alone is sufficient.
   Statement 1: $x^2 > y^2$ could mean $x= -5, y=2$, where $x < y$. Statement 2: $x - y > 0$ directly simplifies to $x > y$.
5. Answer: Statement 1 alone is sufficient.
   Probability is $\frac{ \text{red}}{ \text{total}}$. Statement 1 gives the ratio $3:2$, so probability is $\frac{3}{3+2} = 0.6$. Statement 2 gives a difference but not the total number of marbles.
6. Answer: Statement 2 alone is sufficient.
   Statement 1: $k$ could be 11 or 13 (primes) or 10, 12, 14 (composites). Statement 2 is Wilson's Theorem, which states that a natural number $k > 1$ is a prime number if and only if $(k - 1)! \equiv -1 \pmod{k}$.
7. Answer: Both statements together are sufficient.
   S1: $x-5 = 2$ or $x-5 = -2$, so $x = 7$ or $x = 3$. S2: $x = 7$ or $x = -7$. The only common value is $x = 7$.
8. Answer: Statement 1 alone is sufficient.
   If $4x$ is a perfect square, then $\sqrt{4x} = 2\sqrt{x}$ is an integer. For $2\sqrt{x}$ to be an integer, $\sqrt{x}$ must be an integer or a half-integer (like 0.5). However, since $x$ is an integer, $\sqrt{x}$ must be an integer if its square is an integer.
9. Answer: Statement 1 alone is sufficient.
   The slope $m = \frac{4 - (-4)}{3 - (-3)} = \frac{8}{6} = \frac{4}{3}$. Using $y = mx + b$, $4 = \frac{4}{3}(3) + b \Rightarrow 4 = 4 + b \Rightarrow b = 0$. If the y-intercept is 0, it passes through the origin.
10. Answer: Statement 1 alone is sufficient.
    S1: $y^2$ is always non-negative. For the product to be negative, $x^3$ must be negative, so $x$ is negative. $y^2$ cannot be 0, so $y \neq 0$. However, we don't know the sign of $y$. If $y$ is positive, $xy < 0$. If $y$ is negative, $xy > 0$. Wait, S1 is insufficient. Combining with S2? Still doesn't guarantee the sign of $y$.

1. If \( n \) is an integer, is \( n \) even?

• AStatement 1 alone is sufficient.
• BStatement 2 alone is sufficient.
• CBoth statements together are sufficient.
• DEach statement alone is sufficient.

Frequently Asked Questions

What makes a Data Sufficiency question "hard" on the GRE?

Hard questions often involve constraints that are easy to overlook, such as whether a variable must be an integer, a positive number, or a non-zero value. They frequently use absolute values, inequalities, and number properties like primality to create multiple possible cases.

Should I actually solve for the numerical answer?

No, you should only determine if a unique answer exists. Solving for the exact number often wastes time that could be spent on more complex GRE Reading Passage Questions or other quantitative sections.

Can the two statements in a Data Sufficiency question contradict each other?

On official GRE and GMAT exams, the two statements will never provide contradictory information. If your analysis suggests they do, you should re-check your work for a calculation or logical error.

How do I handle geometry in Data Sufficiency?

Avoid making assumptions based on the visual appearance of a diagram. Use only the provided geometric properties and theorems, such as the Pythagorean theorem, to establish sufficiency.

What is the "C" trap in Data Sufficiency?

The "C" trap occurs when both statements together obviously provide the answer, tempting you to pick "Both together are sufficient" without checking if one statement alone could have answered the question. Always evaluate each statement individually first.