GRE Exam: A Simple Guide To Number Properties - Odds and Evens

An even number is any integer that has 2 as a factor; for example, 4, 28, –12, –6, and so on.
Zero is an even number!
An odd number is simply the opposite: any number that does not have 2 as a factor, for example, 1, 9, –13, and so on.

The GRE will expect you to know the following rules. Though testing numbers is certainly helpful on these questions, you will ultimately save time on the test by committing these rules to memory!

Addition and Subtraction
Property 1: In addition or subtraction, the result will be even if the numbers being added or subtracted are the same (both odd or both even), and the result will be odd if the numbers being added or subtracted are different (one odd and one even).

Multiplication
Property 2: For a product to be even, at least one of the factors must be even.

Let’s look at a sample question that tests these properties:
For this question, indicate all of the answer choices that apply.
If x and y are integers and 3x + 2y is even, then which of the following must be true? (A) x is even (B) x is odd (C) xy is even (D) x + y is even (E) x2 is odd

SOLUTION: Use properties of odds and evens to deduce what must be true about the variables.
3x + 2y = even

Look at each term separately: you don’t know whether 3x is odd or even, since you don’t know whether x is odd or even. However, whether y is odd or even, 2y will always be even since 2y has 2 as a factor.

So think of the equation as:
3x + even = even

From the addition and subtraction rules, you know that even + even = even. Therefore, 3x must be even.

For the product 3x to be even, the term must have an even factor. 3 is not even, so x must be even.

SOLUTION: The correct answer is A and C.