HCF and LCM Explained — KS3 Maths

The highest common factor (HCF) of two numbers is the largest number that divides exactly into both. The lowest common multiple (LCM) is the smallest number that is a multiple of both. Both are found most reliably using prime factor decomposition and a Venn diagram.

What is the highest common factor (HCF)?

The HCF of two numbers is the largest factor that the two numbers share.

Example: HCF of 12 and 18.

Factors of 12: 1, 2, 3, 4, 6, 12

Factors of 18: 1, 2, 3, 6, 9, 18

The largest factor in both lists is 6, so HCF(12, 18) = 6.

What is the lowest common multiple (LCM)?

The LCM of two numbers is the smallest positive number that both numbers divide into exactly.

Example: LCM of 4 and 6.

Multiples of 4: 4, 8, 12, 16, 20 ...

Multiples of 6: 6, 12, 18, 24 ...

The smallest multiple in both lists is 12, so LCM(4, 6) = 12.

Using prime factor trees

For larger numbers, listing factors or multiples becomes slow. The reliable method uses prime factor decomposition — splitting each number into its prime factors.

How to draw a prime factor tree

Write the number at the top.

Split it into any two factors.

Keep splitting each branch until every end is a prime number (circle the primes).

Write the number as a product of its prime factors.

Worked example 1 — prime factor trees for 36 and 60

36: 36 = 4 × 9 = 2 × 2 × 3 × 3 = 2² × 3²

60: 60 = 4 × 15 = 2 × 2 × 3 × 5 = 2² × 3 × 5

Finding HCF and LCM using a Venn diagram

Draw two overlapping circles. Place the prime factors of each number into the diagram:

Shared prime factors go in the overlap (intersection).

Factors belonging only to one number go in the outer regions.

Venn diagram for 36 and 60:

36 only | Both | 60 only 3 | 2², 3 | 5

HCF = product of factors in the overlap:

HCF = 2² × 3 = 4 × 3 = 12

LCM = product of ALL factors in the Venn diagram:

LCM = 3 × 2² × 3 × 5 = 3 × 4 × 3 × 5 = 180

Verify HCF: 36 ÷ 12 = 3 ✓ and 60 ÷ 12 = 5 ✓

Verify LCM: 180 ÷ 36 = 5 ✓ and 180 ÷ 60 = 3 ✓

Worked example 2 — HCF and LCM of 24 and 40

Prime factorisation:

24 = 2³ × 3

40 = 2³ × 5

Venn diagram:

24 only | Both | 40 only 3 | 2³ | 5

HCF = 2³ = 8

LCM = 3 × 2³ × 5 = 3 × 8 × 5 = 120

Check: 24 ÷ 8 = 3 ✓; 40 ÷ 8 = 5 ✓; 120 ÷ 24 = 5 ✓; 120 ÷ 40 = 3 ✓

Worked example 3 — HCF and LCM of 18 and 30

Prime factorisation:

18 = 2 × 3²

30 = 2 × 3 × 5

Venn diagram:

18 only | Both | 30 only 3 | 2, 3 | 5

HCF = 2 × 3 = 6

LCM = 3 × 2 × 3 × 5 = 90

A useful shortcut — the product rule

For any two integers a and b:

HCF(a, b) × LCM(a, b) = a × b

Check with example 3: 6 × 90 = 540 and 18 × 30 = 540 ✓

This rule lets you find the LCM quickly if you know the HCF (or vice versa).

When do HCF and LCM appear in real life?

Scenario | Method needed
Cutting two lengths of ribbon into equal pieces with no waste | HCF (find the longest piece that fits both)
Finding when two buses both arrive at a stop at the same time | LCM (buses run every 12 and 18 minutes → LCM = 36 min)
Simplifying fractions | HCF (divide numerator and denominator by the HCF)
Adding fractions with different denominators | LCM (find the lowest common denominator)

HCF and LCM in the national curriculum

The DfE's KS3 maths programme of study requires pupils to use the concepts and vocabulary of prime numbers, factors, multiples, common factors, and common multiples — including HCF and LCM. BBC Bitesize confirms that prime factor trees and Venn diagrams are the standard KS3 technique, which also forms the foundation for simplifying algebraic expressions at GCSE.

Common mistakes with HCF and LCM

Mistake | Example | Correct approach
Confusing HCF and LCM | Writing LCM(12, 18) = 6 | HCF is the largest SHARED factor; LCM is the smallest SHARED multiple
Not fully completing the prime factor tree | Leaving 4 as a factor instead of splitting to 2 × 2 | Continue splitting until every branch ends in a prime
Forgetting repeated prime factors in the Venn diagram | Placing only one 2 when the overlap needs 2² | Use index notation (e.g. 2³) to keep track of repeated factors

Frequently asked questions

What is the difference between HCF and LCM?

The HCF (highest common factor) is the largest number that divides exactly into both given numbers — it is always less than or equal to the smaller number. The LCM (lowest common multiple) is the smallest number that both given numbers divide into exactly — it is always greater than or equal to the larger number. HCF is used to simplify fractions; LCM is used to find common denominators.

How do I find the HCF and LCM using prime factor trees?

Write the prime factorisation of each number. Draw a Venn diagram with the shared prime factors in the overlap and the unshared factors in the outer regions. The HCF is the product of all factors in the overlap only. The LCM is the product of all factors across the entire Venn diagram.

Can I use the listing method for larger numbers?

The listing method (listing all factors or all multiples) works for small numbers but becomes slow and error-prone for numbers above about 30. Prime factor trees and the Venn diagram method are reliable for any size of number and are the expected method at KS3 and GCSE.

Why is HCF × LCM equal to the product of the two numbers?

Every prime factor of either number appears in the Venn diagram exactly once. The HCF collects the shared factors; the LCM collects all factors. Together, HCF × LCM counts every prime factor from both numbers, which is the same as multiplying the two numbers together. This relationship holds for any two positive integers.

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