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Greatest Common Factor

The highest number that divides exactly into two or more numbers.
It is the "greatest" thing for simplifying fractions!

Greatest Common Factor is made up of the words Factor, Common and Greatest:

What is a "Factor" ?

"Any of the numbers that can be multiplied together to make another number".

It is easier to understand by example:

The factors of 12 are 1,2,3,4,6 and 12 ...

... because 2 × 6 = 12, or 4 × 3 = 12, or 1 × 12 = 12.

What is a "Common Factor" ?

It is a common factor if it is a factor of two or more numbers - it is then "common to" those numbers.

For example, if you work out the factors of two different numbers (say 12 and 30) the common factors are those that are found in both numbers:

The factors of 12 are 1, 2, 3, 4, 6 and 12
The factors of 30 are 1, 2, 3, 5, 6, 10, 15 and 30
 
Notice that 1,2,3 and 6 appear in both lists? So, the common factors of 12 and 30 are: 1, 2, 3 and 6
 

What is the "Greatest Common Factor" ?

It is simply the largest of the common factors. In our previous example, the largest of the common factors is 6, so the Greatest Common Factor is 6

Finding the Greatest Common Factor

You can simply find all factors of both numbers, select the ones that are common to both, and then choose the greatest:

Two Numbers All Factors Common Factors Greatest
Common Factor
Example Simplified
Fraction
9 and 12 9: 1,3,9
12: 1,2,3,4,6,12
1,3 3 9/12 » 3/4
6 and 18 6: 1,2,3,6
18: 1,2,3,6,9,18
1,2,3,6 6 6/18 » 1/3

Or, you can just play around with the factors until you discover it, like this:

Two Numbers Thinking ... Greatest
Common Factor
Example Simplified
Fraction
9 and 12 3 × 3 = 9 and 3 × 4 = 12 3 9/12 » 3/4
24 and 108 2 × 12 = 24, and
9 × 12 = 108
12 24/108 » 2/9
or: 2 × 2 × 2 × 3 = 24, and
2 × 2 × 3 × 3 × 3 = 108
2 × 2 × 3 = 12  

Greatest Common Factor Tool

There is a third method, you can use our Greatest Common Factor Tool to find it automatically.