There are many questions in the GMAT quant section in which the concept of divisibility is important. Therefore, it’s to your advantage to learn the rules in this area.

Sample Question 1

Consider the following question from the Economist GMAT Tutor question database:

If x, y and z are positive integers, and 30x=35y=42z, then which of the following must be divisible by 3?

I) x
II) y
III) z

A) I only
B) II only
C) III only
D) I and III only
E) I, II and III

Using divisibility to solve the problem

A quick way through this question is to realize that 30x is divisible by 3, because it is a product of two integers, one of which is divisible by 3. For a product of integers to be divisible by 3, one or both of the factors must be divisible by 3.

  • Because 30x = 35y, and 30x is divisible by 3, then 35y is also divisible by 3.
  • 35 is not divisible by 3. Therefore y is.
  • 30x is divisible by 3. 30 is divisible by 3, so x may or may not be divisible by 3.
  • In the same way, 42z is divisible by 3. 42 is divisible by 3, so z may or may not be divisible by 3.
  • Therefore, the correct answer is B.

Sample Question 2

Now consider another question involving divisibility from our practice question database:

If a is an even integer and a/18 is an odd integer, which of the following is NOT an even integer?

A) a2 / 54
B) a2 / 12
C) a2 / 9
D) a2 / 6
E) a2 / 2

How to use divisibility to solve this

Plug in a number for a. Let’s take 54, since 54/18 = 3, which fits the restriction that a/18 be an odd integer.

Consider answer choice A.

  • a2 / 54 = 542 / 54, when a = 54
  • 542 / 54 = 54, which is an even integer
  • Eliminate A

Now consider answer choice B.

  • a2 / 12 = 54 * 54 / 12, when a = 54
  • 54 * 54 / 12 = (54 / 12) * 54
  • 54 / 12 is not an integer
  • A non-integer multiplied by an integer will not bring you to an even integer.
  • B is the correct answer

You may be asking yourself: Wait, we are looking for an answer choices that does NOT give us an even integer. Perhaps you were thinking that we need to find an answer choice that gives us an odd integer?

Actually, this question is open to interpretation. NOT an even integer can mean two things:

1. An odd integer
2. Not an integer at all -- a fraction, decimal, or radical

Answer choice B delivers a non-integer, and is the correct answer. (You can go through the other answer choices, but you would not need to.)

If you remember these general principles of divisibility, such questions will become much easier. It is great to have a feeling of confidence when you read a question, rather than a feeling of confusion!