GCSE Maths HCF, LCM & Prime Factorisation Notes
Every whole number can be written as a product of prime factors. From those factors you can quickly find the highest common factor and lowest common multiple.
Key concepts
- Prime factor
- A prime number that divides exactly into a value. 2, 3, 5, 7, 11 … are primes (1 is not prime).
- Product of primes
- Writing a number using only prime factors, e.g. 60 = 2² × 3 × 5, usually found with a factor tree.
- HCF (highest common factor)
- The largest number that divides into two values — multiply the primes they share.
- LCM (lowest common multiple)
- The smallest number both values divide into — multiply each prime the greatest number of times it appears.
Key formulas
Product rule
HCF × LCM = a × b
Handy check: the HCF times the LCM equals the product of the two numbers.
Worked example
Find the HCF and LCM of 24 and 36.
- 1Prime factorise: 24 = 2³ × 3 and 36 = 2² × 3².
- 2HCF = product of shared primes to the lowest power: 2² × 3 = 12.
- 3LCM = each prime to the highest power: 2³ × 3² = 72.
Answer: HCF = 12, LCM = 72
Common mistakes
✗ Confusing HCF and LCM.
✓ HCF is Highest common Factor (a divisor, smaller); LCM is Lowest common Multiple (in the times tables, larger).
✗ Leaving out repeated prime factors, e.g. 24 = 2 × 3.
✓ Include every prime with its power: 24 = 2³ × 3.
✗ Treating 1 as a prime number.
✓ 1 is not prime — the smallest prime is 2.
Quick check
Try these, then reveal the answers.
Write 90 as a product of primes.Reveal answer
2 × 3² × 5
HCF of 18 and 30?Reveal answer
6
LCM of 4 and 6?Reveal answer
12
Exam tips
- Draw a factor tree and circle the primes so nothing is missed.
- A Venn diagram makes HCF (the overlap) and LCM (everything) easy to read off.
- 'Product of prime factors' means index form — write 2³ × 3, not 2 × 2 × 2 × 3, unless asked.
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Frequently asked questions
How do I find the HCF from prime factors?+
Multiply together the prime factors that appear in both numbers, using the lowest power of each shared prime.
What is prime factorisation used for?+
It underpins HCF, LCM, simplifying surds and fractions, so it is one of the most reused skills on the paper.
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