Great Common Factor

Greatest Common Factor

Our math class is learning about GCF. Here is some information that will help.

The highest number that divides exactly into two or more numbers.
It is the "greatest" thing for simplifying fraction (putting fractions in lowest terms)!

Greatest Common Factor is made up of three words

• Greatest,
• Common and
• Factor

What is a "Factor" ? Factors are the numbers you multiply together to get another number:

2 X 3 = 6 2 and 3 are factors

Sometimes we want to find ALL the factors of a number:

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" ? Let us say you have worked out the factors of two or more 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

Then the common factors are those that are found in both numbers:

• 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

It is a common factor when it is a factor of two or more numbers. (It is then "common to" those numbers.) Example: What are the common factors of 15, 30 and 105?

The factors of 15 are 1, 3, 5, and 15

The factors of 30 are 1, 2, 3, 5, 6, 10, 15 and 30

The factors of 105 are 1, 3, 5, 7, 15, 21, 35 and 105

The factors that are common to all three numbers are 1, 3, 5 and 15

In other words, the common factors of 15, 30 and 105 are 1, 3, 5 and 15

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 15, so the Greatest Common Factor of 15, 30 and 105 is 15

The "Greatest Common Factor" is the largest of the common factors of two or more numbers

Why is this Useful? One of the most useful things is when we want to simplify a fraction:

Example: How could we simplify ¹²/₃₀ ?

At the top we found that the Common Factors of 12 and 30 were 1, 2, 3 and 6, and so the Greatest Common Factor is 6.

This means that the largest number we can divide both 12 and 30 evenly by is 6, like this:

12 ÷ 6 = 2

30 ÷ 6 = 5

The Greatest Common Factor of 12 and 30 is 6. And so ¹²/₃₀ can be simplified to ²/₅

Finding the Greatest Common Factor Here are three ways:

1. You can:

• find all factors of both numbers (All Factors Tool to help you),
• then select the ones that are common to both, and
• then choose the greatest.

Example:

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

And another example:

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

2. You can find the prime factors and combine the common ones together:

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

3. And sometimes you can just play around with the factors until you discover it:

Two Numbers	Thinking ...	Greatest Common Factor	Example Simplified Fraction
9 and 12	3 × 3 = 9 and 3 × 4 = 12	3	9/12 » 3/4

But in that case you had better be careful you have found the greatest common factor.

Math Rules!!