“Odd/Even” seems like a very basic concept. All of us first learned about odd and even numbers when we were very young. However, the GMAT finds interesting and sometimes unexpected ways to test your understanding of these number properties.
Let’s take a look at an example problem:
X is an odd integer and Y is an even integer, and neither X nor Y is equal to zero. Which of the following expressions must represent an odd integer?
A. 4(X + Y)
B. 2X + Y
C. 2Y + X
D. X ÷ Y
E. Y ÷ X
There are a number of ways you could attempt to solve the problem. We could try plugging in example values for X and Y. However, there are some shortcuts.
For starters, it’s very easy to eliminate at least one of the answer choices. Just by inspection, choice A could NEVER by odd. Here’s why: even though we don’t know what X or Y equals, we do know that the sum of X + Y is being multiplied by 4. And since 4 itself is an even number, 4(X + Y) has to also be an even number. So, remember that when finding multiplying two or more integers together, if at least one of the integers is even, then the overall product of the integers will be even.
The shortcut that we used to eliminate choice A can also help us to better understand the other choices.
In Choice B, 2X must be an even number, because multiplying the even number 2 times X has to give an even result. In Choice C, 2Y must be an even number for the same reason.
So we can restate Choices B and C as follows:
B. Even number + Even Number (because we were told that Y itself is even)
C. Even number + Odd Number (because we were told that X itself is odd).
The restated choices help us to finish solving the problem very quickly. Choice B is eliminated because the pattern “Even + Even = Even” is always true. Meanwhile, Choice C has to be the answer we're looking for, because the pattern “Even + Odd = Odd” is always true.
Even though we already know the answer is C, let’s look at Choices D and E to see what we can learn from them.
In Choice D, X divided by Y means we have an odd integer divided by an even integer. Notice that the result has to be a NON-integer. For example, 7 divided by 2 equals 3.5. Unless provided with a special definition within the problem, we consider non-integers to be neither odd nor even.
In Choice E, Y divided by X means an even integer divided by an odd integer. Here the result could be either of two possibilities. Let’s use numbers to illustrate. If we had 6 divided by 3, the answer would be 2, which is an even integer. If we had 8 divided by 3, the answer would be approximately 2.67, which is a non-integer. Generalizing, when we divide an even integer by an odd integer, we will get either an even integer or a non-integer.
Addition, Subtraction, and Multiplication of odd and/or even numbers always follow predictable patterns.
However, division is not as predictable.
Here’s a summary of basic operations with odd and even integers:
Addition and Subtraction
Even + Even = Even
Odd + Odd = Even
Even + Odd = Odd
Even - Even = Even
Odd - Odd = Even
Even - Odd = Odd
Odd - Even = Odd
Notice that with addition and subtraction, pairing two numbers of the same parity (evenness or oddness) will give an even answer, while pairing two numbers of different parity will give an odd answer.
Even x Even = Even
Even x Odd = Even
Odd x Odd = Odd
Notice that when multiplying, “even always wins.”
You can see here that division is not as predictable as the other operations.
Even ÷ Even = Even or Odd: Example: 6 ÷ 2 = 3, but 8 ÷ 2 = 4
Odd ÷ Odd = Odd or Non-integer. For example: 21 ÷ 3 = 7, but 21 ÷ 5 = 4.2
Even ÷ Odd = Even or Non-integer. For example: 20 ÷ 5 = 4, but 20 ÷ 9 = 2.222...
Since the operations with Addition, Subtraction, and Multiplication are predictable, if you forget the general rules, you can quickly plug in a couple of example numbers to verify what the general pattern is.
Just remember that Division is a bit trickier.
Good luck on the GMAT!