Break numbers into prime factors, then use them to find the highest common factor and lowest common multiple. It is on Foundation and Higher, aimed at grade 4. Free questions with answers are on this page. Worked examples show each line before you try the questions.
A prime-factor tree (or repeated division) writes a number as a product of primes: 84=22×3×7. That unique fingerprint is how you find HCF and LCM of larger numbers without listing twenty factors. HCF: take each prime that appears in both, to the lowest power. LCM: take each prime that appears in either, to the highest power.
A Venn diagram of the primes makes the 'both / only A / only B' split visible. Product of HCF and LCM equals the product of the two numbers — a useful check. Worded HCF ('largest tiles that fit') and LCM ('first time they coincide') are this algebra in a sentence.
A factor tree (or repeated division) writes a number as a product of primes. HCF takes the primes in both lists, using the lowest power. LCM takes every prime that appears, using the highest power. Bus and alarm questions are LCMs; 'largest pack that divides both' is HCF.
Objectives
A prime has no factors other than 1 and itself.
Split a composite number into a pair of factors, then split those, until every end is prime.
The highest common factor uses only the primes that appear in both lists, at the lower power.
Key points
Write a number as a product of its prime factors using a factor tree or repeated division.
Use index notation such as 23×32×5.
Find the HCF by taking the primes that appear in both lists (lowest power of each).
Find the LCM by taking every prime that appears anywhere (highest power of each).
Worked examples
Example 1
What is the highest common factor of 8 and 20?
3 marks
8 = 2³ and 20 = 2² × 5.
Take the lower power of each shared prime: 2² = 4.
The highest common factor is 4.
Answer: 4
Common slips
Slip: Write 8 and 20 as products of primes, and use every prime from both lists. The HCF only uses primes that appear in both numbers.
Slip: Take the higher power of every prime: 2³ × 5 = 40. The HCF takes the lower power of each shared prime, not the higher power of every prime.
Slip: The highest common factor is 40. That is the lowest common multiple, not the highest common factor.
Example 2
What is the lowest common multiple of 8 and 20?
3 marks
8 = 2³ and 20 = 2² × 5.
Take the higher power of every prime: 2³ × 5 = 40.
The lowest common multiple is 40.
Answer: 40
Common slips
Slip: Write 8 and 20 as products of primes, and use only the shared primes. The LCM needs every prime from either number.
Slip: Take the lower power of each shared prime: 2² = 4. The LCM takes the higher power of every prime, not the lower power of the shared ones.
Slip: The lowest common multiple is 4. That is the highest common factor. The LCM is a multiple of both numbers, so it cannot be smaller than them.
Example 3
What is the highest common factor of 12 and 18?
3 marks
12 = 2² × 3 and 18 = 2 × 3².
Take the lower power of each shared prime: 2 × 3 = 6.
The highest common factor is 6.
Answer: 6
Common slips
Slip: Write 12 and 18 as products of primes, and use every prime from both lists. The HCF only uses primes that appear in both numbers.
Slip: Take the higher power of every prime: 2² × 3² = 36. The HCF takes the lower power of each shared prime, not the higher power of every prime.
Slip: The highest common factor is 36. That is the lowest common multiple, not the highest common factor.
Questions and answers
GCSE Maths prime factors, hcf & lcm questions and answers. Try each one, then open the answer. Easy questions come first.
Question 1 Easy2 marks
Find the highest common factor of 12 and 18.
Show answer
Answer: 6
List the factors, or use prime factors.
The highest one in both lists is the HCF.
M1 — List the factors, or use prime factors.
A1 — Answer 6
Question 2 Easy2 marks
Find the lowest common multiple of 6 and 8.
Show answer
Answer: 24
The LCM is the smallest number both divide into.
The value is 24.
M1 — The LCM is the smallest number both divide into.
A1 — Answer 24
Question 3 Medium2 marks
Bus A runs every 6 minutes and bus B every 8 minutes. They leave together. After how many minutes are they together again?
Show answer
Answer: 24
This is the LCM of the two intervals.
The value is 24.
M1 — This is the LCM of the two intervals.
A1 — Answer 24
Question 4 Medium2 marks
Two ropes are 24 cm and 36 cm. They are cut into equal pieces as long as possible, with none left over. How long is each piece?
Show answer
Answer: 12
The piece length is the HCF.
The value is 12.
M1 — The piece length is the HCF.
A1 — Answer 12
Question 5 Hard2 marks
Two whole numbers are in the ratio 2:3 and their highest common factor is 4. What is the larger number?
Show answer
Answer: 12
The HCF is one part of the ratio.
The larger number is 3 parts.
M1 — The HCF is one part of the ratio.
A1 — Answer 12
Question 6 Hard2 marks
A bell rings every 4 minutes and another every 6 minutes. How many times do they ring together in 60 minutes, not counting the start?
Show answer
Answer: 5
They ring together every 12 minutes.
60 ÷ 12 is the number of times after the start.
M1 — They ring together every 12 minutes.
A1 — Answer 5
Question 7 Exam-style3 marks
Two whole numbers are in the ratio 2:3 and their highest common factor is 5. What is the larger number?
Show answer
Answer: 15
The HCF is one part of the ratio.
The larger number is 3 parts.
M1 — The HCF is one part of the ratio.
M1 — The larger number is 3 parts.
A1 — Answer 15
Common mistakes
Most lost marks on this topic are one slip.
Including 1 as a prime.
1 is not prime.
Using the higher power in the HCF.
The HCF takes the lower shared power.
Multiplying the two numbers and calling that the LCM.
That product is a common multiple, but not always the lowest.
Spot the slip
Which of these is prime?
2
1
9
Show the slip
2 2 has exactly two factors, 1 and itself.
Exam tips
Be ready to do this without a calculator. Paper 1 is the non-calculator paper.
If the question says “show that”, the result is given. The marks are for the steps that reach it.
The method is the same on AQA-style and Edexcel-style papers.
Show your factor tree or division ladder — it earns a mark even if you copy a number wrongly at the end.
On a calculator paper, the fraction button can help: enter 12084 and simplify; the number you divided by is the HCF.
Printable worksheet
The same questions are in a PDF you can print. Answers start on a later page.