Prime Factorisation, HCF and LCM –

Prime factorisation, HCF (highest common factor), and LCM (lowest common multiple) form a core Number topic tested on every GCSE Maths exam board at both Foundation and Higher tier. These skills are needed for simplifying fractions, solving problems involving repeated events, and working with ratios. This page explains what prime factorisation is, shows you how to use factor trees and Venn diagrams to find the HCF and LCM, provides worked examples at both tiers, and gives you practice questions to cement your understanding. If you are planning your revision schedule, our how to revise GCSE Maths guide can help you organise your time.

What Is Prime Factorisation?

A prime number is a number greater than 1 that has exactly two factors: 1 and itself. The first few primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29.

Prime factorisation means writing a number as a product of its prime factors. Every whole number greater than 1 can be expressed uniquely as a product of primes — this is known as the Fundamental Theorem of Arithmetic.

For example: 60 = 2² × 3 × 5.

Key Definitions

• Highest Common Factor (HCF): The largest number that divides exactly into two or more numbers. Also called the greatest common divisor.
• Lowest Common Multiple (LCM): The smallest number that is a multiple of two or more numbers.

Key Relationships

Step-by-Step Method

Finding the Prime Factorisation (Factor Tree)

1. Write the number at the top of the tree.
2. Split it into any two factors (one of which should be prime if possible).
3. Circle any prime factors — these are the "leaves" of the tree.
4. Continue splitting non-prime factors until every branch ends in a prime.
5. Write the result using index notation, listing primes in ascending order.

Finding the Prime Factorisation (Repeated Division)

1. Divide the number by the smallest prime that goes into it (usually 2).
2. Write the quotient below.
3. Repeat, dividing by the smallest possible prime each time.
4. Continue until the quotient is 1.
5. The prime factorisation is the product of all the primes you divided by.

Finding the HCF Using Prime Factorisation

1. Write the prime factorisation of each number.
2. Identify the prime factors that appear in both factorisations.
3. For each common prime, take the lowest power.
4. Multiply these together to get the HCF.

Finding the LCM Using Prime Factorisation

1. Write the prime factorisation of each number.
2. For each prime that appears in either factorisation, take the highest power.
3. Multiply these together to get the LCM.

Using a Venn Diagram

1. Draw two overlapping circles, one for each number.
2. Place the common prime factors (with their lowest powers) in the overlap — this represents the HCF.
3. Place the remaining prime factors in the non-overlapping parts.
4. The HCF is the product of everything in the overlap.
5. The LCM is the product of everything in the entire Venn diagram.

Worked Example 1 — Foundation Level

Question: Find the HCF and LCM of 36 and 90.

Working:

Step 1 — Prime factorisations:

• 36 = 2² × 3²
• 90 = 2 × 3² × 5

Step 2 — HCF: take the lowest power of each common prime.

• 2 appears in both: lowest power is 2¹ = 2
• 3 appears in both: lowest power is 3² = 9
• HCF = 2 × 9 = 18

Step 3 — LCM: take the highest power of each prime.

• 2: highest power is 2² = 4
• 3: highest power is 3² = 9
• 5: highest power is 5¹ = 5
• LCM = 4 × 9 × 5 = 180

Answer: HCF = 18, LCM = 180

Worked Example 2 — Higher Level

Question: Two lighthouses flash at intervals of 12 seconds and 18 seconds respectively. They both flash at the same time. After how many seconds will they next flash at the same time?

Working:

Step 1 — This is an LCM problem — we need the smallest time that is a multiple of both 12 and 18.

Step 2 — Prime factorisations:

• 12 = 2² × 3
• 18 = 2 × 3²

Step 3 — LCM: take the highest power of each prime.

• 2²: from 12
• 3²: from 18
• LCM = 4 × 9 = 36

Answer: They will next flash at the same time after 36 seconds.

Common Mistakes

• Confusing HCF and LCM. HCF uses the lowest powers of common primes; LCM uses the highest powers of all primes. A helpful mnemonic: HCF = Highest is wrong, it is the Highest Common Factor but found using the lowest shared powers.
• Not breaking a number down completely to primes. 12 = 4 × 3 is not fully factorised because 4 is not prime. You must write 12 = 2² × 3.
• Forgetting to include primes that only appear in one number when finding the LCM. For example, if one number has a factor of 5 and the other does not, the 5 must still be included in the LCM.
• Writing the factorisation without index notation. Examiners expect you to use powers: write 2³ × 3 × 5, not 2 × 2 × 2 × 3 × 5.

Exam Tips

• Learn to draw factor trees quickly. Practise until you can factorise numbers up to 200 in under a minute.
• Venn diagrams are very popular in exam mark schemes. If a question says "use prime factorisation to find the HCF and LCM", a Venn diagram is the clearest way to present your working.
• Know when a problem requires HCF vs LCM. HCF questions usually involve splitting things into equal groups or finding the largest shared quantity. LCM questions involve repeated events or finding when things coincide. Our formulas guide lists the key relationships.
• Verify your answer using HCF × LCM = product of the two numbers. For example, 18 × 180 = 3240, and 36 × 90 = 3240. They match, so the answer is correct.

Practice Questions

Q1 (Foundation): Write 84 as a product of its prime factors. Give your answer in index notation.

Q2 (Foundation): Find the HCF of 48 and 72.

Q3 (Higher): Bus A leaves the station every 14 minutes. Bus B leaves every 20 minutes. Both leave at 09:00. When do they next leave at the same time?

Practise prime factorisation, HCF, and LCM with unlimited AI-generated questions. Start for free and track your progress.

Related Topics

• Fractions
• Powers and Roots
• Indices and Index Laws
• Order of Operations (BIDMAS)

Summary

• Prime factorisation expresses a number as a product of prime numbers using index notation.
• Use a factor tree or repeated division to find the prime factorisation.
• HCF: multiply the lowest powers of the common prime factors.
• LCM: multiply the highest powers of all prime factors that appear in either number.
• A Venn diagram is the clearest way to organise your working for HCF and LCM.
• HCF × LCM = product of the two original numbers — use this to check your answers.
• LCM problems involve coinciding events; HCF problems involve equal grouping.