Highest Common Factor (HCF)
The highest common factor (HCF) is the greatest whole number that divides exactly into two or more numbers. It is especially useful for solving real-world problems involving dividing items into the largest equal groups, or cutting materials into the longest possible equal lengths with nothing left over. Jump to the questions
Practise now
Worksheet preview and key skills
Worksheet preview
This worksheet helps you practise finding the highest common factor (HCF) of different sets of numbers. You will work through simple pairs before progressing to larger numbers and worded problems.
The self-marking activity includes questions with two numbers, three numbers, and real-life scenarios to test your understanding of what the highest common factor actually represents.
What you’ll practise
- Finding the HCF of simple number pairs
- Recognising when one number is an exact factor of the other
- Understanding that when numbers share no other factors, their HCF is 1
- Finding the HCF of three numbers
- Using the highest common factor to solve equal-length and identical-group word problems
Use the interactive worksheet below, or read the Topic guide for the method and worked example.
Find the highest common factor of each pair or set of numbers. The HCF is the greatest whole number that divides exactly into every number.
Topic guide
What is a factor?
A factor is a whole number that divides exactly into another number with no remainder. For example, the factors of 12 are 1, 2, 3, 4, 6 and 12.
What is a common factor?
When you compare the factors of two different numbers, any factors that appear in both lists are called common factors. For example, if we compare 12 and 18, their common factors are 1, 2, 3 and 6.
What is the highest common factor?
The highest common factor (HCF) is simply the greatest number that appears in both lists. Looking at our common factors of 12 and 18 (which were 1, 2, 3 and 6), the largest is 6. So, the HCF of 12 and 18 is 6.
A method using factor lists
For manageable numbers, the most reliable method is to list out all the factors of each number in pairs, then find the largest match.
Worked example: Find the HCF of 18 and 30
First, list all the factors of 18 in pairs:
- 1 and 18
- 2 and 9
- 3 and 6
Next, list all the factors of 30 in pairs:
- 1 and 30
- 2 and 15
- 3 and 10
- 5 and 6
The common factors in both lists are 1, 2, 3 and 6. The largest of these is 6, so the HCF of 18 and 30 is 6.
Finding the HCF of three numbers
The method is exactly the same for three numbers. You list the factors for all three numbers and look for the largest number that appears in every list. For example, to find the HCF of 24, 36 and 60, you would find that 12 is the largest number dividing exactly into all three.
The special case: one number is a factor of the other
Sometimes, the smaller number divides exactly into the larger number. For example, with 14 and 56, 14 divides exactly into 56. In this case, the smaller number itself (14) is the highest common factor.
What does it mean when the HCF is 1?
Every whole number has 1 as a factor. If two numbers do not share any other common factors (for example, 15 and 22), their highest common factor is exactly 1. They are not "0", because 0 is not a factor of any non-zero number.
Recognising HCF word problems
In worded problems, you might not be told explicitly to "find the HCF". You must recognise when it is required. Look out for situations involving:
- Cutting materials into the longest equal lengths with nothing left over.
- Dividing different quantities into the greatest number of identical groups with nothing left over.
Common mistakes
- Choosing a number that is a factor of only one of the numbers.
- Selecting a common factor, but stopping before you find the highest one.
- Forgetting that a number itself is always one of its own factors.
- Saying the HCF is 0 when the only common factor is 1.
- Confusing the highest common factor (HCF) with the lowest common multiple (LCM).