Greatest Common Factor
The highest number that divides exactly into two or more numbers.
It is the "greatest" thing for simplifying fractions!
Let's start with an Example ...
Greatest Common Factor of 12 and 16
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So ... what is a "Factor" ?
Factors are the numbers you multiply together to get another number:
A number can have many factors:
Factors of 12 are 1, 2, 3, 4, 6 and 12 ...
...
because 2 × 6 = 12, or 4 × 3 = 12, or 1 × 12 = 12.
(Read how to find All the Factors of a Number. In our case we don't need the negative ones.)
What is a "Common Factor" ?
Let us say you have worked out the factors of two numbers:
Example: Factors of 12 and 30
| Factors of 12 are 1, 2, 3, 4, 6 and 12 |
| Factors of 30 are 1, 2, 3, 5, 6, 10, 15 and 30 |
Then the common factors are those that are found in both lists:
- 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.)
Here is another example with three numbers:
Example: The common factors of 15, 30 and 105
| Factors of 15 are 1, 3, 5, and 15 |
| Factors of 30 are 1, 2, 3, 5, 6, 10, 15 and 30 |
| 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 ?
Earlier we found that the Common Factors of 12 and 30 were 1, 2, 3 and 6, and so the Greatest Common Factor is 6.
So the largest number we can divide both 12 and 30 evenly by is 6, like this:
| ÷ 6 | ||
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| 12 | = | 2 |
| 30 | 5 | |
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| ÷ 6 | ||
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 (I have an All Factors Calculator to help you),
- then select the ones that are common to both, and
- then choose the greatest.
Example:
| Two Numbers | 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 | 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.
Greatest Common Factor Calculator
There is another easy method, you can use our Greatest Common Factor Calculator to find it automatically.
Other Names
The "Greatest Common Factor" is often abbreviated to "GCF", and is also known as:
- the "Greatest Common Divisor (GCD)", or
- the "Highest Common Factor (HCF)"