# 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

- Find all the
**Factors**of each number, - Circle the
**Common**factors, - Choose the
**Greatest**of those

## So ... what is a "Factor" ?

Factors are numbers we can 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" ?

Say we 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.

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 can we simplify ** \frac{12}{30} **?

Earlier we found that the Common Factors of 12 and 30 are 1, 2, 3 and 6, and so the **Greatest Common Factor is 6**.

So the **largest** number we can divide both 12 and 30 exactly by is **6**, like this:

÷ 6 | ||

\frac{12}{30} | = | \frac{2}{5} |

÷ 6 |

The Greatest Common Factor of 12 and 30 is **6**.

And so ** \frac{12}{30} ** can be simplified to ** \frac{2}{5}**

## Finding the Greatest Common Factor

Here are three ways:

**1.** We can:

- find all
**factors**of both numbers (use the All Factors Calculator), - then find the ones that are
**common**to both, and - then choose the
**greatest**.

Example:

Two Numbers | Factors | Common Factors | Greatest Common Factor |
Example SimplifiedFraction |
---|---|---|---|---|

9 and 12 | 9: 1,3,912: 1,2,3,4,6,12 |
1,3 | 3 |
\frac{9}{12} = \frac{3}{4} |

And another example:

Two Numbers | Factors | Common Factors | Greatest Common Factor |
Example SimplifiedFraction |
---|---|---|---|---|

6 and 18 | 6: 1,2,3,618: 1,2,3,6,9,18 |
1,2,3,6 | 6 |
\frac{6}{18} = \frac{1}{3} |

**2**. Or we can find the prime factors and combine the common ones together:

Two Numbers | Thinking ... | Greatest Common Factor |
Example SimplifiedFraction |
---|---|---|---|

24 and 108 | 2 × 2 × 2 × 3 = 24, and2 × 2 × 3 × 3 × 3 = 108 |
2 × 2 × 3 = 12 |
\frac{24}{108} = \frac{2}{9} |

**3.** Or sometimes we can just **play around** with the factors until we discover it:

Two Numbers | Thinking ... | Greatest Common Factor |
Example SimplifiedFraction |
---|---|---|---|

9 and 12 | 3 × 3 = 9 and 3 × 4 = 12 |
3 |
\frac{9}{12} = \frac{3}{4} |

But in that case we must check that we have found the **greatest** common factor.

## Greatest Common Factor Calculator

OK, there is also a really *easy* method: we can use the 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)"