"If X and Y Are Different Prime Numbers, Each Greater Than 2..." — GMAT® Worked Solution

Source: Official Guide for GMAT® Review, 11th Edition

If $x$ and $y$ are different prime numbers, each greater than 2, which of the following must be true?

I. $x + y \neq 91$

II. $x - y$ is an even integer

III. $\dfrac{x}{y}$ is not an integer

(A) II only

(B) I and II only

(C) I and III only

(D) II and III only

(E) I, II, and III

Try it before reading on.

Step 1: Understand What "Must Be True" Means

"Must be true" means the statement is true 100% of the time, with no exceptions. If we can find even one case where a statement is false, it's not "must be true" and we don't select it.

The bar for proving a statement is very high. We need to be sure it holds for every possible pair of different primes greater than 2.

Step 2: Write the Constraints

Before evaluating any statement, write down what we know:

• $x$ and $y$ are prime numbers
• $x$ and $y$ are different
• $x > 2$ and $y > 2$

Since 2 is the only even prime number, every prime greater than 2 is odd. So both $x$ and $y$ are odd.

Step 3: Evaluate the Easiest Statement First

Looking at all three statements, II looks easiest because it only involves even/odd reasoning. We don't need to think about large primes or specific values.

Statement II: $x - y$ is an even integer.

Both $x$ and $y$ are odd, since they're primes greater than 2. The difference of any two odd numbers is always even. So Statement II must be true.

Eliminate any answer choice that doesn't include II:

• (A) II only — still possible
• (B) I and II only — still possible
• (C) I and III only — eliminate
• (D) II and III only — still possible
• (E) I, II, and III — still possible

Four options remain. Not a huge reduction, but it's progress, and it gives us leverage if we need to guess later.

Step 4: Evaluate the Next Easiest Statement

Statement III looks easier than Statement I because it doesn't require thinking about large primes.

Statement III: $\dfrac{x}{y}$ is not an integer.

Prime numbers are only divisible by 1 and themselves. Since $x$ and $y$ are different primes, $y$ is not a factor of $x$. The only factors of $x$ are 1 and $x$, and $y$ is neither of those because $y$ is a different prime greater than 2.

So $\dfrac{x}{y}$ cannot be an integer. Statement III must be true.

Eliminate any answer choice that doesn't include III:

• (A) II only — eliminate
• (B) I and II only — eliminate
• (D) II and III only — still possible
• (E) I, II, and III — still possible

Down to two options. Even if we can't evaluate Statement I, we have a 50% chance instead of 20%.

Step 5: Evaluate Statement I

Statement I: $x + y \neq 91$.

This one seems harder because 91 is a specific number, and we might need to think about large primes. But we already know something important: both $x$ and $y$ are odd.

The sum of two odd numbers is always even. 91 is odd. So $x + y$ is always even, which means it can never equal 91.

Statement I must be true.

The answer is (E).

All three statements must be true.

Step 6: The Systematic Listing Alternative

If the even/odd reasoning didn't click for Statement I right away, there's another approach: list the smallest possible pairs systematically and look for a pattern.

Start with the smallest primes that satisfy the constraints and add them up:

The pattern is clear after 4 or 5 pairs: the sums are all even. You don't need to list every possible pair of primes up to 91 to see it. The first few pairs reveal the pattern, and the even/odd rule confirms it.

This listing technique is simple, but it's high-leverage. When a property seems hard to prove directly, listing the smallest possible values in a systematic way will often reveal a pattern within 4 or 5 entries. The key word is "systematic": start with the smallest possible values and move up one increment at a time, rather than picking random pairs.

Why This Problem Matters

About 30% of test takers miss this problem, and 21% pick (D): II and III only. That's a heavily concentrated wrong answer, which points to a specific error rather than random guessing.

The most likely cause is forgetting the constraint that $x$ and $y$ are greater than 2. Without that constraint, 2 is a valid prime, and 2 plus 89 equals 91, which would make Statement I seem false. The problem explicitly says "greater than 2" to prevent this, but under time pressure, constraints get dropped from short-term memory.

Another possibility: seeing the "$\neq$" symbol and reading it as "$=$". If you misread "does not equal 91" as "equals 91," you'd look for a pair of primes that sums to 91, find 2 and 89, and conclude that Statement I doesn't have to be true. Same wrong answer, different path.

The fix: write all constraints at the top of your scratchwork before evaluating any statement. "Greater than 2" is the constraint that makes the entire problem work. Without it written down, you're relying on memory, which is unreliable under exam conditions. The same habit that prevents the exponent inequality error (double-checking transcription) prevents this error (writing constraints). Both problems test execution, not knowledge. The math is simple. The setup is where points are lost.

Want the full strategy behind this problem? Read: GMAT® Quant: Three Systems for Exponents, Primes, and Percent Change

From Episode 27 of Real GMAT® Problems (The GMAT® Strategy Podcast).

Ready for the next one? "If Candy Bars That Regularly Sell for 40 Cents Each Are on Sale at Two for 75 Cents..." — GMAT® Worked Solution.