Finding The Greatest Common Factor

Greatest Common Factor of two numbers is the result of two numbers being factored into their smaller factors individually and of all the numbers that are factors, the one that is greatest is the greatest common factor:

Symbol: ( ) means greatest common factor.

For instance, (6, 8) means the greatest common factor of 6 and 8.

Example: (6, 8) is,

First, find all the factors of 6.  The factors of 6 are all the numbers that when multiplied by another number give us 6.  Those are, 1, 2, 3, 6.  This is so because:

1*6=6 and 2*3=6

So you see that each number when multiplied by another number gives us 6.

Similarly,

Second, Find the factors of 8.
Those are, 1, 2, 4, 8.

So, now, when we look at the factors of 6 and 8
for 6: 1, 2, 3, 6
for 8: 1, 2, 4, 8

We see that 1, and 2, both appear in the factors of both six and 8:

Now, of the factors that appear in both numbers, that is 1 and 2, the greatest one is 2. *

We use greatest instead of greater because when we speak of bigger numbers, there might be more than just two numbers that are common factors.

A second way that works very well with larger numbers is to break our two numbers into their prime factors. For example;

180 can be broken down into (2²)(3²)(5¹), or 2 squared times 3 squared times 5 to the first. When looking at two different number we merely have to find the largest power of each prime factor that appears in both numbers. So, if I wanted to find the greatest common factor of 180 and 234, I would break down both of these numbers.

180 = (2²)(3²)(5¹)
234 = (2¹)(3²)(13¹).

So our common factors are 2 and 3, and the greatest common power of 2 is one and of 3 is two, so the greatest common factor is 2¹*3² = 18.

This way is very nice when working with large numbers with lots of factors.