Guides for parents · 1 November 2026
HCF and LCM: the quick way with prime factors
The highest common factor (HCF) is the biggest number that divides into both numbers. The lowest common multiple (LCM) is the smallest number that both divide into. One pair of factor trees gives you both.
The listing method
For small numbers, listing is fine and builds understanding. Take 24 and 36.
Factors of 24: 1 2 3 4 6 8 12 24 Factors of 36: 1 2 3 4 6 9 12 18 36 Common factors: 1 2 3 4 6 12 HCF = 12 Multiples of 24: 24 48 72 96 ... Multiples of 36: 36 72 108 ... First in both: 72 LCM = 72
Listing gets slow once the numbers pass 100, and it is easy to miss one. That is where prime factors help.
The prime factor method
Write both numbers as products of primes (see prime factors) and line up the matching primes.
24 = 2 × 2 × 2 × 3 36 = 2 × 2 × 3 × 3 Shared: 2 × 2 × 3 = 12 HCF Shared × leftovers: 12 × 2 × 3 = 72 LCM
A Venn diagram shows the same thing: the shared primes go in the middle. The HCF is the middle multiplied together; the LCM is everything in the diagram multiplied together.
In power form, take the smaller power of each shared prime for the HCF (2² × 3 = 12) and the larger power of every prime for the LCM (2³ × 3² = 72).
A check that catches most slips
- The HCF can't be bigger than the smaller number. The LCM can't be smaller than the bigger number.
- For two numbers, HCF × LCM equals the two numbers multiplied: 12 × 72 = 864 = 24 × 36.
Which one does the question want?
Word problems rarely say "HCF" or "LCM". In England, common factors and common multiples are part of Year 6, and these problems are common in 11+ papers.
- Things happening together again (buses, lights, bells, laps) → LCM. Buses leave every 12 and every 18 minutes, both at 9:00. LCM(12, 18) = 36, so they next leave together at 9:36.
- Cutting or sharing into the biggest equal pieces (ribbons, tiles, teams, party bags) → HCF. Ribbons of 48 cm and 60 cm are cut into equal pieces, as long as possible, with nothing left over: HCF(48, 60) = 12 cm, giving 4 + 5 = 9 pieces.
A quick test: if the answer should be bigger than the numbers in the question, it's the LCM. If it should be smaller, it's the HCF.
Common mistakes
- Multiplying the two numbers for the LCM. 24 × 36 = 864 is a common multiple, but not the lowest one.
- Swapping the two. Use the size check above.
- Forgetting that the HCF can be 1. 8 and 15 share no prime, so their HCF is 1 and their LCM is 8 × 15 = 120.
Where it turns up next
Simplifying a fraction divides the top and bottom by their HCF: 24/36 becomes 2/3 after dividing both by 12. Adding fractions uses the LCM of the denominators as the common denominator.