GCSE MATHS • HIGHER TIER

HCF and LCM Made Simple

Learn how to find the Highest Common Factor and Lowest Common Multiple using lists and prime factor decomposition.

Clear method

Exam practice

Worked proofs

HCF and LCM in Four Moves

1 Identify

Decide whether the problem requires a common factor or a common multiple.

2 List or Decompose

List factors/multiples or write each number as a product of prime factors.

3 Compare

Find the shared factors or multiples, or compare prime powers.

4 Select

Choose the highest shared factor for HCF or the lowest shared multiple for LCM.

What You’ll Learn

Understand HCF

Know that HCF is the largest factor shared by two or more numbers.

Understand LCM

Know that LCM is the smallest multiple shared by two or more numbers.

Use Listing Methods

Find HCF using factors and LCM using multiples.

Use Prime Factors

Apply prime factor decomposition to larger numbers.

Solve Real-Life Problems

Use HCF for equal groups and LCM for repeating events.

Key Rules

HCF Uses Factors

HCF is the largest factor shared by all the given numbers.

LCM Uses Multiples

LCM is the smallest multiple shared by all the given numbers.

Lowest Powers for HCF

When using prime factors, select the lowest shared powers.

Highest Powers for LCM

When using prime factors, select the highest powers required.

Worked Proof: HCF and LCM

Step-by-Step Method

1. Decide whether the question requires HCF or LCM.
2. List the factors or multiples of each number.
3. Find the factors or multiples common to all numbers.
4. Choose the highest common factor for HCF.
5. Choose the lowest common multiple for LCM.

More Worked Examples

1

Find the HCF of 20 and 30

Factors of 20 :

1, 2, 4, 5, 10, 20
Factors of 30 :
1, 2, 3, 5, 6, 10, 15, 30
Common factors :
1, 2, 5, 10

HCF = 10

2

Find the LCM of 5 and 8 .

Multiples of 5 :

5, 10, 15, 20, 25, 30, 35, 40
Multiples of 8 :
8, 16, 24, 32, 40

LCM = 40

3

Find the HCF of 24 and 36

24 = 2³ × 3

36 = 2² × 3²
Take the lowest powers .

HCF = 2² × 3 = 12

4

Find the LCM of 24 and 36 .

24 = 2³ × 3

36 = 2² × 3²
Take the highest powers .

LCM = 2³ × 3² = 72

Look for words such as largest equal groups when deciding whether HCF is needed.
Look for repeating events or when things happen together when deciding whether LCM is needed.
For small numbers, listing factors or multiples can be quick and reliable.
For larger numbers, prime factor decomposition is often more efficient.
Use the lowest shared prime powers for HCF.
Use the highest required prime powers for LCM.