GCSE MATHS • HIGHER TIER

Factors, Multiples and Prime Numbers Made Simple

Build strong GCSE Number skills by identifying factors, multiples, prime numbers, prime factors, HCF and LCM.

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Factors, Multiples and Prime Numbers in Four Moves

1

Identify

Decide whether the question is about factors, multiples or prime numbers.

2

List

Generate factor pairs, multiples or possible divisors carefully.

3

Break Down

Use prime factor decomposition or common lists when needed.

4

Compare

Find common factors or multiples to determine HCF or LCM.

What You’ll Learn

Understand Factors

Know which numbers divide exactly into another number.

Understand Multiples

Generate multiples by multiplying by integers.

Recognise Prime Numbers

Identify numbers with exactly two factors.

Use Prime Factor Decomposition

Write numbers as products of prime factors.

Find HCF and LCM

Find the highest common factor and lowest common multiple.

Key Rules

Factors Divide Exactly

A factor divides into a number without leaving a remainder.

Multiples Come from Multiplication

Multiples are found by multiplying a number by positive integers.

Prime Numbers Have Two Factors

A prime number has exactly two factors: 1 and itself.

1 Is Not Prime

The number 1 has only one factor, so it is not prime.

Worked Proof: Factors, Multiples and Prime Numbers

Step-by-Step Method

  • 1. Decide whether the question involves factors, multiples or prime numbers.
  • 2. List the factor pairs, multiples or possible divisors.
  • 3. Use a factor tree to break numbers into prime factors if needed.
  • 4. Compare the lists to find the common factors or multiples.
  • 5. Choose the largest common factor for the HCF or the smallest common multiple for the LCM.

More Worked Examples

1

List all factors of 20

Factor pairs :

1 × 20
2 × 10
4 × 5
Answer: 1, 2, 4, 5, 10, 20

2

List the first six multiples of 7.

Answer: 7, 14, 21, 28, 35, 42

3

Write 36 as a product of prime factors.

36 = 2 × 2 × 3 × 3

= 2² × 3²

4

Find the LCM of 4 and 6

Multiples of 4 :

4, 8, 12, 16 ...
Multiples of 6 :
6, 12, 18 ...
Answer = 12
  • Common GCSE Mistakes
  • Definitions: Do not confuse factors with multiples.
  • Prime numbers: Remember that 1 is not prime.
  • Factor pairs: Check systematically so no factors are missed.
  • Factor trees: Continue until every branch ends in a prime number.
  • Exam Tips
  • Use factor pairs to list factors systematically.
  • Write several multiples when searching for a common multiple.
  • Remember that a prime number has exactly two factors.
  • Use a factor tree until only prime factors remain.
  • For HCF, choose the largest shared factor.
  • For LCM, choose the smallest shared multiple.

Skills Used in Factors, Multiples and Prime Numbers

Factors

Multiples

Prime Numbers

Factor Pairs

Prime Factor Decomposition

Factor Trees

Highest Common Factor

Lowest Common Multiple

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Frequently Asked Questions

A factor divides exactly into another number without leaving a remainder.

A multiple is produced by multiplying a number by an integer.

A prime number has exactly two factors: 1 and itself.

No. The source states that 1 is not a prime number.

The HCF is the largest factor shared by two numbers.

The LCM is the smallest multiple shared by two numbers.