Highest common factor

Tier: #Foundation #Higher

🔗What you need to know first

How to

The highest common factor (HCF) of two or more numbers is the largest number that divides exactly into all of them.

Method 1 — list factors: Factors of 12: 1, 2, 3, 4, 6, 12 Factors of 18: 1, 2, 3, 6, 9, 18 HCF = 6

Method 2 — prime factor decomposition: $$12 = 2^2 \times 3 \qquad 18 = 2 \times 3^2$$ Multiply shared prime factors using the lowest power of each: $$\text{HCF}(12, 18) = 2^1 \times 3^1 = 6$$

Using a Venn diagram: place prime factors of each number in a Venn diagram — the HCF is the product of factors in the intersection.

Why it matters: used to simplify fractions (divide numerator and denominator by HCF) and to simplify ratios.

Common error: confusing HCF with LCM — HCF divides into the numbers, LCM is a multiple of the numbers.

Common topics that go alongside

Prime factor decomposition Simplifying fractions Lowest Common Multiple

❓Questions to practise

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📝Past paper questions

2025 Jun 1H GCSE Q1 (2 marks)

2024 Jun 3H GCSE Q1 (2 marks)

2023 Nov 1H GCSE Q3 (2 marks)

2023 Jun 3H GCSE Q5 (1 mark)

2022 Nov 1H GCSE Q8 (2 marks)

2021 Nov 2H GCSE Q2 (2 marks)

2019 Jun 1H GCSE Q3 (2 marks)

2018 Jun 2H GCSE Q2 (1 mark)

💬What the examiners say

"There was a significant number who found correct prime factors, and entered these into a Venn diagram, but were then unsure how to use this to complete the question."
"Effective use was made of Venn diagrams, with students generally scoring at least one of the two marks and often both."

⬆️How you can quickly improve

Use factor trees for both numbers, draw a Venn diagram, then the HCF is the product of the prime factors in the overlapping section only.
Label clearly at the top of your working whether you're finding HCF or LCM before any calculation — don't let them blur into each other.
Verify the HCF by checking it divides evenly into both original numbers.

ℹ️Calculator tricks