## Arithmetic on the GRE

# GRE integer review

Arithmetic: adding and subtracting and all those multiplication tables you memorised in primary school. It was the first math that you learned and will (in all likelihood) be the math you use most throughout your life. You know arithmetic, right?

Well, what if you were asked the difference between a number and an integer? Or the distinct properties of zero? Or what makes a number a "real" number? The GRE will test your knowledge of these (supposedly) basic principles in tricky and subtle ways. No matter how good a math student you were (or are!), every student's prep for the GRE should include a review of arithmetic.

Let's begin by discussing integers.

**What is an integer?**

Integers are what your teachers most likely called "whole numbers." Examples of integers are: 1, 2, 3, 47 and 419, as well as their negative counterparts. Zero is also an integer. Fractions, radicals and numbers with decimals attached are *not*** **integers.

Much like every square is a rectangle but not all rectangles are squares, all integers are numbers but not all numbers are integers.

Integers have several rules regarding what happens to them when they are manipulated:

Any integer added to, subtracted from or multiplied by another integer will result in an answer of an integer (an integer divided by another may result in a non-integer answer)

Multiplying two positive integers will

*always*result in a positive integerMultiplying two negative integers will

*always*result in a positive integerMultiplying a positive integer by a negative one will

*always*result in a negative integer

## The GRE also features questions about factors and multiples.

**Factors** are numbers that divide into your integer cleanly; **multiples** are numbers that your integer divides into.

Here’s an example:

The factors of 42 are 1, 2, 3, 6, 7, 14, 21, and 42. (All numbers are their own factors and multiples.)

1 x 42 = 42

2 x 21 = 42

3 x 14 = 42

6 x 7 = 42

The multiples of 42 are 42, 84, 126, etc.

A neat trick to remember which is which: think of “FM radio.” “F” (factor) comes first, and is therefore all the numbers *before *(or smaller) than your integer. “M” (multiple) comes second, and is therefore *after* (or bigger) than your integer.

Are there negative factors and multiples? Of course! Just as 6 x 7 = 42, so too does -6 x -7.

The **Least Common Multiple (LCM)*** *is something you will surely see on the GRE. Simply put, the LCM is the smallest number that two integers divide into cleanly. For example, 20 and 30 share multiples of 60, 120, 180, 240, etc. The smallest of these numbers—60—is the LCM.

The **Greatest Common Factor (GCF) **is the highest factor shared by two integers. Using 20 and 30 again, we see that 20's factors are 1, 2, 4, 5, 10 and 20. The factors of 30 are 1, 2, 3, 5, 6, 10, 15 and 30. The largest number that both integers share as a factor is 10, therefore, 10 is the GCF for 20 and 30.

**Even and Odd**

Integers can be either even or odd. Even integers are divisible by 2; odd numbers are not. Zero (despite not being able to be divided) is considered to be even.

Here are some quick rules about the calculation of even and odd integers:

The product of two even integers is an even integer

The product of two odd integers is an odd integer

The product of an even integer and an odd integer is an even integer

The sum of two even integers is an even integer

The sum of two odd integers is an even integer

The sum of an even integer and an odd integer is an odd integer

**Prime Numbers**

A prime number is an integer that only has two distinct divisors: itself and 1. Prime numbers include 2, 3, 5, 7, 11, 13, 17 and 19. Four (4), for example, is not a prime number because, besides itself and 1, it has a divisor of 2.

Some more quick rules:

One (1) is

*not*a prime number (a prime number must have two distinctTwo (2) is the

*only*even prime numberA prime number

*must*be an integerA prime number

*must*be positive

This overview of integers should demonstrate that this seemingly basic math area warrants attention during your test prep studies. The GRE is chock full of math questions that rely on the rules of integers, so prospective test-takers should plan to spend time strengthening their foundation.