Greatest Common Factor (gcf)

The Greatest Common Factor (gcf), also sometimes called the Greatest Common Divisor (gcd), is the largest positive integer that divides two non-zero integers without remainder.

The Greatest Common Factor of two number, say a and b, is usually written in the form gcf(a,b). Here are some examples:

gcf(9,18) = 9

gcf(4,18) = 2

gcf(9, 28) = 1 *

* Note in the case when the gcf of two numbers is 1 the numbers are called coprime or relatively prime.

How to calculate the gcf:

We know from the Fundamental Theorem of Arithmetic that every number can be broken down to a product of prime numbers. This is called the number's unique prime factorization. One we find this for the two number we wish to calculate the gcf of, we simply take the numbers that match exactly in each number's unique factorization and multiply them together. Here's a simple example:

gcf(120, 156)

Well, first we find the unique factorization of each number:

120 = 2 * 2 * 2 * 2 * 3 * 5
156 = 2 * 2 * 3 * 13

The numbers that have an exact match in the other's factorization (noted above in red) are multiplied together:

2 * 2* 3 = 12

So,

gcf(120,156) = 12