# GRE Arithmetic - Primes and Divisibility

A **prime number** is a number that can be divided evenly by only itself and 1. For example, 3 is a prime number because it can only be divided by 3 and 1 without a remainder, but 6 is not a prime number, because 6 can be divided by 1, 2, 3, and 6. Alternatively, a **composite** number is a number that has more than 2 divisors. The numbers 0 and 1 are NOT prime numbers, by definition. The number 2 is special because it is the only even number that is a prime number. GRE questions will often ask you to count the number of primes in a range of numbers.

Test Tip |
For example, the first 15 positive prime numbers, starting with 2, are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43 and 47. Should you memorize these? NO! But you should be able to look at a number and determine if it is not a prime number. If the number can be divided by 2, 3, 4, 5, ..., etc, or by any number other than itself and 1, then it is NOT prime. |

If there are two numbers and the first number can be divided evenly by the second number, then the first number is **divisible** by the second number. For example, 25 is divisible by 5, because you can divide 25 by 5 with no remainder. In this example, 5 is also called a **factor** of 25, because it can divide into 25 evenly. So here is a sample question:

What is the sum of

the divisors of 36?

The divisors of 36 are 1, 2, 3, 4, 6, 9, 12, 18 and 36, and so their sum is 1 + 2 + 3 + 4 + 6 + 9 + 12 + 18 + 36 = 91.

In addition, **prime factors** are numbers which are both factors of a number and which are prime. Of 36, 2 and 3 are the only prime factors; 4, 6, 9, 12 and 18 are not prime factors of 36, because these are not prime numbers. So, here is a sample question that might appear on the GRE exam:

What are the prime factors of 54?

First, you must determine the factors of 54, which are 1, 2, 3, 6, 9, 18, 27 and 54, and which of those are prime? Only 2 and 3 are prime numbers -- remember that 1, by definition, is NOT a prime number.

As you can see, the best way to answer such "factor" or "prime factor" questions requires you to enumerate all possible factors or primes of a number. Of course, don't expect to be asked on the GRE to calculate the factors or primes of large numbers, because that would be too much tedious work. However, any number less than 100 is fair game for a "factor" or "prime factor" question.