To find out the answer to ‘what is the highest common factor,’ we need to know what a factor is. So we also need to know what a common factor is and how to discover the highest common factor.

Firstly, a factor is a number that divides into another number exactly without leaving a remainder.

Secondly, a common factor is a factor that is shared by two or more numbers. For instance:

- A common factor of 8 and 10 is 2, as 2 is a factor of 8, and 2 is a factor of 10.

Finally, the highest common factor (HCF) is found by calculating all common factors of two numbers and choosing the largest numerical value.

Simply put, the highest common factor of two, or more, numbers is the highest whole number that is a factor of both.

**Examples of finding factors**

A factor is a number that is multiplied to get another number.

For example, 6 and 2 are both factors of 18 because they can be multiplied by another number to reach 18.

- 6 × 3 = 18
- 2 × 9 = 18

You can see by these two equations that 3 and 9 must also be factors of 18.

So now we know that one number can have many factors. 18 has 1, 2, 3, 6, 9, and 18.

All these can be multiplied by another number in the list to get 18.

**How do I find the highest common factor?**

Typically, for problems where you find the highest common factor, you will be given two numbers.

There are a few methods that you can use to find the highest common factor:

**Listing method**

To find the highest common factor of these numbers, list every factor of both numbers in their respective lists, then compare the two lists of factors to find the highest number they have in common. For example:

*Find the highest common factor of 15 and 60*

Elements of 15: 1, 3, 5, 15

Aspects of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60

*The highest common factor of 15 and 60 is 15.*

You may also have to find the highest common factor of more than two numbers. For example:

*Find the highest common factor of 18, 120, and 42*

Factors of 18: 1, 2, 3, 6, 9, 18

Factors of 120: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120

Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42

The highest common factor of these three numbers is 6.

**Prime factorization**

The second method you can use to find the highest common factor is prime factorization. To use this method, you must carry out the following steps:

Step 1: List the prime factors of each of your given numbers and note the ones in common.

Step 2: Multiply the common prime factors together, and then find the highest common factor of those numbers.

Let’s break this down through an example:

Find the highest common factor of 40 and 60.

Step 1: List the prime factors of each of those numbers.

40 = 2 × 2 × 2 × 5

60 = 2 × 2 × 3 × 5

Now that we have the prime factors of each number listed above, we can easily identify the common prime factors.

The common prime factors of 40 and 60 are 2, 2, and 5.

Step 2: Multiply the common prime factors together.

2 × 2 × 5 = 20

So, the highest common factor of 40 and 60 is 20.

**Division method**

You can also use the long division method to know the highest common factor of two numbers.

You need to divide the larger number by the smaller number, then divide the smaller number by the remainder until there is no longer one.

For example, 504 ÷ 318 gives us 1 with a remainder of 186. Next, we’ll divide 318 by the rest of 186, which provides us with 1 with a remainder of 132.

We continue dividing 186 by the remainder of 132, which gives us a rest of 54.

By dividing 132 by 54, we get 2 with a remainder of 24. Finally, we divide 54 by 24 to get 2 with a rest of 6.

24 divides perfectly into 6, so our final answer is 6.