Take the GRE Challenge: Problem Solving

Problem Solving questions test various math concepts, including percentages, symbolism, special triangles, median, range, probability, and more. You'll also have to read graphs, interpret data, and solve math problems—including word problems.

So if today were Test Day, how would you do?

Try this Problem Solving question and find out.

If the integer a is a multiple of 6, the integer b is a multiple of 3, and a > b > 0, then which of the following integers must a2 - b2 be a multiple of?

A. 6
B. 8
C. 9
D. 10
E. 12

And the answer is...


We can use the strategy of Picking Numbers to solve this question. With the method of Picking Numbers, all 4 incorrect answer choices must be eliminated because sometimes one or more incorrect answer choices will work for the particular values that we select. Let's let a = 6 and b = 3. Then a2 - b2 = 62 - 32 = 36 - 9 = 27. Looking at the answer choices, 27 is not a multiple of 6, 8, 10, or 12, which are the numbers in answer choices (A), (B), (D), and (E), respectively. So choices (A), (B), (D), and (E) can be eliminated. Now that all 4 incorrect answer choices have been eliminated, we know that choice (C) must be correct.

Just to check, we see that 27 is a multiple of 9, which is to be expected since we have concluded that choice (C) must be correct.

Here is an algebraic solution to this question:

Since a is a multiple of 6, we can say that a = 6N1, where N1 is a positive integer. Since b is a multiple of 3, we can say that b = 3N2 , where N2 is a positive integer. Then a2 - b2 = (6N1 )2 - (3N2 )2 = 36N12 - 9N22 = 9(4N12 - N22). Since N1 is an integer, 4N12 = 4 x N1 x N1 is an integer. Since N2 is an integer, N2 2 = N2 x N2 is an integer. Since 4N12 and N22 are both integers, 4N12 - N22 is an integer. So a2 - b2 is 9 multiplied by the integer 4N12 - N22 . Thus, a2 - b2 must be a multiple of 9. Choice (C) is correct.