HCF and LCM –

Finding the Highest Common Factor (HCF) and Lowest Common Multiple (LCM) is a fundamental Number topic tested on every GCSE Maths exam board. These skills underpin work with fractions, ratios, and algebraic fractions, so mastering them is essential for both Foundation and Higher tiers.

What Are HCF and LCM?

The Highest Common Factor (HCF) of two numbers is the largest number that divides exactly into both of them. For example, the HCF of 12 and 18 is 6, because 6 is the biggest number that goes into both 12 and 18.

The Lowest Common Multiple (LCM) of two numbers is the smallest number that is a multiple of both. For example, the LCM of 4 and 6 is 12, because 12 is the smallest number that appears in both the 4 times table and the 6 times table.

There are three main methods: listing, prime factor decomposition, and the Venn diagram method. For larger numbers, prime factorisation with a Venn diagram is by far the most efficient and is the method examiners prefer to see.

Key Formulas

Step-by-Step Method

1. Write each number as a product of its prime factors (prime factor decomposition).
2. Draw a Venn diagram with two overlapping circles.
3. Place the shared prime factors in the overlap.
4. Place the remaining prime factors in the outer parts of each circle.
5. HCF = multiply all the numbers in the overlap.
6. LCM = multiply all the numbers in the entire Venn diagram.

Worked Example 1 — Foundation Level

Question: Find the HCF and LCM of 24 and 36.

Working:

Step 1 — Prime factorise: 24 = 2 × 2 × 2 × 3 = 2³ × 3, and 36 = 2 × 2 × 3 × 3 = 2² × 3².

Step 2 — Venn diagram: shared factors are 2 × 2 × 3 (= 12). Remaining for 24: one extra 2. Remaining for 36: one extra 3.

Step 3 — HCF = 2 × 2 × 3 = 12.

Step 4 — LCM = 2 × 2 × 2 × 3 × 3 = 72.

Answer: HCF = 12, LCM = 72

Worked Example 2 — Higher Level

Question: Find the HCF and LCM of 60 and 90.

Working:

Step 1 — 60 = 2² × 3 × 5 and 90 = 2 × 3² × 5.

Step 2 — Shared factors: 2 × 3 × 5. Remaining for 60: one extra 2. Remaining for 90: one extra 3.

Step 3 — HCF = 2 × 3 × 5 = 30.

Step 4 — LCM = 2 × 2 × 3 × 3 × 5 = 180.

Step 5 — Check: 30 × 180 = 5,400. And 60 × 90 = 5,400. Correct.

Answer: HCF = 30, LCM = 180

Worked Example 3 — Exam Style

Question: Two buses leave the station at 9:00 am. Bus A returns every 12 minutes and Bus B returns every 20 minutes. At what time will both buses next be at the station together? (3 marks)

Working:

Step 1 — Find the LCM of 12 and 20. Prime factorise: 12 = 2² × 3 and 20 = 2² × 5.

Step 2 — LCM = 2² × 3 × 5 = 60 minutes.

Step 3 — 60 minutes after 9:00 am is 10:00 am.

Answer: 10:00 am

Common Mistakes

• Confusing HCF and LCM. The HCF is always smaller than or equal to both numbers. The LCM is always greater than or equal to both. If your HCF is bigger than one of the numbers, you have mixed them up.
• Missing a prime factor in the decomposition. If you stop the factor tree too early (e.g. leaving a 6 instead of breaking it into 2 × 3), your HCF and LCM will be wrong.
• Forgetting to use the HCF × LCM check. HCF × LCM should equal the product of the two numbers. Use this to verify your answer quickly.

Exam Tips

• Draw the Venn diagram neatly in the exam — examiners often award a mark specifically for a correct Venn diagram.
• When a question says "use prime factorisation," you must show the factor trees or repeated division. The listing method will not earn full marks.
• Bus timetable and alarm clock problems are classic LCM contexts. Light pattern problems (two lights flashing at different intervals) also use LCM.

Practice Questions

Q1 (Foundation): Find the HCF of 28 and 42.

Q2 (Foundation): Find the LCM of 6 and 10.

Q3 (Higher): Find the HCF and LCM of 84 and 120.

Practise HCF and LCM questions with instant feedback — completely free on GCSEMathsAI.

Related Topics

• Prime Factorisation, HCF and LCM
• Fractions
• Adding and Subtracting Fractions

Summary

• The HCF is the largest factor common to both numbers.
• The LCM is the smallest multiple common to both numbers.
• Use prime factorisation and a Venn diagram for the most reliable method.
• HCF = product of the shared prime factors (the overlap).
• LCM = product of all prime factors in the Venn diagram.
• Check your answer: HCF × LCM = the product of the two original numbers.