Curriculum fit: GCSE Maths Number (Foundation and Higher, including Venn diagram method). US: Grade 6 greatest common factor and least common multiple.
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What Is It?
Every whole number above 1 can be written as a product of prime numbers in exactly one way. This is called its prime factorisation. A factor tree is a simple way to find it.
The highest common factor (HCF, called GCF in the US) is the largest number that divides into two numbers. The lowest common multiple (LCM) is the smallest number that both divide into. Prime factorisation makes both easy to find.
Key Formulas and Ideas
- HCF: multiply the prime factors the two numbers share (using the lower power of each).
- LCM: multiply all prime factors that appear, using the higher power of each.
- For two numbers a and b: HCF × LCM = a × b.
Worked Examples
Prime Factorisation
Write 60 as a product of prime factors.
- 60 = 6 × 10.
- 6 = 2 × 3 and 10 = 2 × 5.
- Collect the primes: 2 × 2 × 3 × 5.
✓ Answer 60 = 2² × 3 × 5
HCF
Find the HCF of 36 and 48.
- 36 = 2² × 3² and 48 = 2⁴ × 3.
- Take the lower power of each shared prime: 2² and 3.
- Multiply: 4 × 3 = 12.
✓ Answer 12
LCM
Find the LCM of 36 and 48.
- Take the higher power of each prime: 2⁴ and 3².
- Multiply: 16 × 9 = 144.
✓ Answer 144
Common Mistakes to Avoid
- Stopping a factor tree before every branch ends in a prime.
- Mixing up HCF and LCM. The HCF is never larger than either number and the LCM is never smaller.
- Using the higher power for the HCF instead of the lower.
- Leaving the answer as a list instead of writing it with powers, such as 2² × 3.
Practice Questions
Frequently Asked Questions
When would I use LCM in real life?
To find when two repeating events line up, such as buses that leave every 12 and 18 minutes.
What if two numbers share no prime factors?
Their HCF is 1, and their LCM is the product of the two numbers.
Keep Going With These Guides
Factors, Multiples and Prime Numbers Explained
Learn what factors, multiples and prime numbers are, how to list them and how to test whether a number is prime, with worked examples.
KS3GCSEMiddle SchoolHigh SchoolVenn Diagrams and Set Notation: How to Read and Use Them
Use sets, Venn diagrams and the symbols for union, intersection and complement to solve problems and find probabilities, with examples.
KS3GCSEMiddle SchoolHigh SchoolIndices and Exponent Rules: The Laws of Indices Explained
Learn the index laws for multiplying, dividing and raising powers, plus zero and negative indices. UK indices and US exponent rules, with examples.
GCSEHigh SchoolSources and Review
- DfE: GCSE mathematics subject content and assessment objectives
- Common Core State Standards: Mathematics (high school)
Reviewed by StudyMorrow editorial review on 5 Nov 2026.