Fractions Fundamentals: Greatest Common Factor (GCF)

When you factor (do factoring on) two numbers, the largest number in common between them is the greatest common factor. Our first example will be the numbers 2 and 4.

The factors of 2 are 1 and 2:

2 ÷ 1 = 2
2 ÷ 2 = 1

The factors of 4 are 1, 2, and 4:

4 ÷ 1 = 4
4 ÷ 2 = 2
4 ÷ 4 = 1

The greatest number in common between
1, 2
and
1, 2, 4
is 2.

Now let's look for the greatest common factor between the numbers 10 and 25.

The factors of 10 are 1, 2, 5, and 10:

10 ÷ 1 = 10
10 ÷ 2 = 5
10 ÷ 5 = 2
10 ÷ 10 = 1

The factors of 25 are 1, 2, 5, and 25:

25 ÷ 1 = 25
25 ÷ 5 = 5
25 ÷ 25 = 1

The greatest number in common between
1, 2, 5, 10
and
1, 5, 25
is 5.

Here's another example: the numbers 12 and 72.

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

12 ÷ 1 = 12
12 ÷ 2 = 6
12 ÷ 3 = 4
12 ÷ 4 = 3
12 ÷ 6 = 2
12 ÷ 12 = 1

The factors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, and 72:

72 ÷ 1 = 72
72 ÷ 2 = 36
72 ÷ 3 = 24
72 ÷ 4 = 18
72 ÷ 6 = 12
72 ÷ 8 = 9
72 ÷ 9 = 8
72 ÷ 12 = 6
72 ÷ 18 = 4
72 ÷ 24 = 3
72 ÷ 72 = 1

The greatest number in common between
1, 2, 3, 4, 6, 12
and
1, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
is 12.

Sometimes, two numbers won't have a number that is common between them except for the number 1. An example of this are the numbers 12 and 19:

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

12 ÷ 1 = 12
12 ÷ 2 = 6
12 ÷ 3 = 4
12 ÷ 4 = 3
12 ÷ 6 = 2
12 ÷ 12 = 1

The factors of 19 are 1 and 19 because 19 is a prime number:

19 ÷ 1 = 19
19 ÷ 19 = 1

The greatest number in common between
1, 2, 3, 4, 6, 12
and
1, 19
is 1.