What is the fastest way to finding the highest common factor (HCF) between two numbers, and how can you check your answer?

The quickest way to find the highest common factor between two numbers is to start by identifying all the factors of the smallest number first, then with this list, starting from the biggest factors, see if each number divides into the larger number without leaving a remainder.

As you progress down through the list of factors (from the smaller number) when you find a number that divides into the larger number you can stop as you have found the
**
highest common factor of both numbers
**
.

For example, with the numbers 9 and 12 the highest common factor is 3.

## HCF Of 9 And 12

Here is a step by step guide demonstrating how to get the highest common factor between the numbers 9 and 12:

With the two numbers, identify which is the smallest: 9 or 12?

9 is the smaller number.

With the smaller number 9, find all the factors of this number.

Start by inserting into your factor list both the number 1 and 9.

Working factor list contains 1 and 9

Increment the factor by 1: 1 + 1 = 2

Can 2 be divided into 9 without leaving a remainder?

No! 2 divided into

As you have reached (or are about to eclipse) the square root of 9 you can stop here.

You now have all the factors of 9 being 1, 3 and 9.

Starting from the end of your factor list, see if each number can be divided into the larger number without any remainder.

Can 9 be divided into 12 without leaving a remainder?

No! 9 divided into 12 leaves a remainder of 3.

Therefore, you need to move to the next factor of the smallest number in the list.

Can 3 be divided into 12 without leaving a remainder?

Yes! 3 can be divided into 12 without leaving a remainder.

Therefore, 3 is the
**
highest common factor
**
of 9 and 12.

As you can see from the example above the step-by-step process in working with the factors of the smallest number can help to quickly achieve the answer of what the highest common factor is for both numbers.

## How To Check Your Answer

Is there a way you can check your answer if you’ve used the approach above? Yes, there is!

An alternative approach to finding the highest common factor between two numbers is to find the factors of the
*
largest number first
*
and then see if each factor can be divided into the smallest number. To help quicken the pace in checking your answer you can remove any factors from the larger number that are greater than the smaller number.

Here’s how this approach would work using the same numbers above:

With the two numbers, identify which is the
*
largest
*
of the two numbers: 9 or 12? 12 is the larger of the two numbers.

With this number, find all its factors.

Start by inserting into your factor list both the number 1 and 12.

Working factor list contains 1 and 12

Increment the factor by 1: 1 + 1 = 2

Can 2 be divided into 12 without leaving a remainder?

Yes! Therefore, add both 2 and the quotient 6 into the factor list.

Working factor list contains 1, 2, 6 and 12

Increment the factor by 1: 2 + 1 = 3

Can 3 be divided into 12 without leaving a remainder?

Yes! Therefore, add both 3 and the quotient 4 into the factor list.

Working factor list contains 1, 2, 3, 4, 6 and 12

As you have reached (or are about to eclipse) the square root of 12 you can stop here.

You now have all the factors of 12 being 1, 2, 3, 4, 6 and 12.

Remove the numbers from this list that are
*
larger
*
than the smaller number 9.

This will mean your factor list now only contains: (1, 2, 3, 4, 6)

Starting from the end of the list, see if each number can be divided into the smaller number without any remainder.

Can 6 be divided into 9 without leaving a remainder?

No! 6 divided into 9 leaves a remainder of 3.

Therefore, you need to move to the next factor in the list.

Can 4 be divided into 9 without leaving a remainder?

No! 4 divided into 9 leaves a remainder of 1.

Therefore, you need to move to the next factor in the list.

Can 3 be divided into 9 without leaving a remainder?

Yes! 3 can be divided into 9 without leaving a remainder.

Therefore, 3 is the
**
highest common factor
**
of 9 and 12.

As you can see both approaches achieve the same answer, which helps to give you confidence with your original answer.

## Summary

The highest common factor (HCF) between 2 numbers is the largest number that can be divided into both numbers without leaving a remainder. The HCF of 9 and 12 is 3.

Finding the highest common factor is a skill used quite frequently in arithmetic, especially when reducing fractions.