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GCSE Maths · Number
Prime Factors, HCF and LCM GCSE Questions and Worked Answers
The HCF is the largest positive whole number that divides each number exactly. The LCM is the smallest positive whole number in each number’s times table. Prime factors give a systematic way to find both.
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Start with the meaning
What you need to know about prime factors, hcf and lcm
Eighteen counters can make three groups of six with nothing left over. That makes 6 a factor of 18: it divides 18 exactly. Making more groups of six produces 6, 12, 18, 24 and so on. These are multiples of 6. A factor fits inside an amount; a multiple is made from whole copies of it.
See the idea first
A shared divisor is different from a shared times-table answer
For positive whole numbers, ‘common’ means appearing in every list. HCF means highest common factor: the greatest shared factor. LCM means lowest common multiple: the least shared positive multiple. Zero is not a candidate for the LCM.
Factors of 181, 2, 3, 6, 9, 18every number here divides 18 exactly→
Factors of 301, 2, 3, 5, 6, 10, 15, 30the shared factors are 1, 2, 3 and 6→
Greatest shared factorHCF = 66 is the largest common divisor
| Number | First multiples |
|---|---|
| 6 | 6, 12, 18, 24, 30, 36… |
| 8 | 8, 16, 24, 32, 40… |
| First match | LCM of 6 and 8 is 24 |
From problem to method
Typical problems you need to be able to solve
These are the main kinds of question in this topic. Each card first states the problem, then explains how to solve it.
Write a number as prime factors
What the problem asks: Break 60 into a product of prime numbers.
How to solve it: A prime is a whole number greater than 1 with exactly two positive factors: 1 and itself. Split 60 into 6 × 10, then 2 × 3 × 2 × 5. The final factors are all prime, so 60 = 2² × 3 × 5.
Make the greatest number of identical packs
What the problem asks: Share 18 pens and 30 pencils into identical packs, using everything.
How to solve it: The number of packs must divide both counts. Their HCF is 6, so make six packs with three pens and five pencils in each.
Find when repeating events coincide
What the problem asks: Two lights flash every 6 and 8 seconds, starting together. When do they next flash together?
How to solve it: A common flash time must be in both times tables. The first positive match, 24 seconds, is the LCM.
Find HCF and LCM from prime factors
What the problem asks: Find the HCF and LCM of 36 and 54.
How to solve it: 36 = 2² × 3² and 54 = 2 × 3³. Their shared copies give HCF = 2 × 3² = 18. Enough copies to contain both give LCM = 2² × 3³ = 108. A power such as 3² means two copies of 3 multiplied.
A reliable routine
Find HCF and LCM using prime factors
Use this method for positive whole numbers, particularly when listing factors or multiples is long. Each prime factorisation lists the building blocks of a number; a common divisor needs shared blocks, while a common multiple must contain all the required blocks.
- Split each number into factors until all end factors are prime. Do not stop at 4, 6 or 9.
- Write repeated prime factors using powers.
- For HCF, take the smaller exponent for every shared prime.
- For LCM, take the larger exponent for every prime appearing in either number.
- Check that the HCF divides both numbers and the LCM is divisible by both.
Check: 1 is not prime. If two numbers have no common prime factors, their HCF is 1, not 0.
Fully worked
Prime Factors, HCF and LCM GCSE worked examples
Each solution explains the clue, why the method fits and how to check the result.
Example 1
Build a prime factorisation
Question
Write 84 as a product of prime factors in index form.
Split into manageable factors, then keep splitting composite factors.
Every final factor is prime. Multiplying back gives 84.
Example 2
Find a common divisor
Question
Find the HCF of 36 and 54.
Both contain one 2 and two 3s.
Tip: the extra 2 in 36 and extra 3 in 54 are not shared.
Example 3
Find a common multiple
Question
Find the LCM of 36 and 54.
Use 36 = 2² × 3² and 54 = 2 × 3³. A multiple of both needs two 2s and three 3s.
Check: 108 ÷ 36 = 3 and 108 ÷ 54 = 2.
Example 4
Make identical packs
Question
A club has 42 badges and 70 stickers. It makes the greatest possible number of identical packs using everything. Find the number of packs and the contents of each.
The pack count must divide both totals.
Make 14 packs. Each contains 42 ÷ 14 = 3 badges and 70 ÷ 14 = 5 stickers. Tip: distinguish number of packs from items per pack.
Example 5
Repeating events
Question
Two bells ring every 12 and 18 minutes. They ring together at 10:05. When do they next ring together?
The next coincidence is 36 minutes after 10:05, at 10:41. The HCF would not give a time in both ringing schedules.
Example 6
A missing number
Question
Two positive integers have HCF 6 and LCM 90. One number is 18. Find the other.
For two positive integers, their product equals HCF × LCM: for each prime, HCF takes the smaller power and LCM the larger. Their product therefore contains exactly the same prime copies as the two original numbers multiplied. Let the other number be n.
Check: HCF(18, 30) = 6 and LCM(18, 30) = 90. This product rule is for two numbers.
10 original questions · total 22 marks
Prime Factors, HCF and LCM GCSE exam-style questions
Try each question before opening its fully worked answer. The difficulty rises through the set.
Before you startAllow about 27 minutes · Show your working. Give exact answers unless a question asks you to round, and include units where needed. · answers start collapsed
List factors
List all positive factors of 20.
Show worked answer
Use factor pairs 1 × 20, 2 × 10 and 4 × 5. The factors are 1, 2, 4, 5, 10 and 20.
Identify primes
Which of 1, 2, 9, 11 and 15 are prime?
Show worked answer
2 and 11 are prime. Each has exactly two positive factors. 1 has only one; 9 = 3 × 3 and 15 = 3 × 5 are composite.
Prime factorisation
Write 90 as a product of prime factors.
Show worked answer
HCF
Find the HCF of 24 and 40.
Show worked answer
Both have three copies of 2.
LCM
Find the LCM of 24 and 40.
Show worked answer
Include 2³, 3 and 5.
Three numbers
Find the HCF and LCM of 12, 18 and 30.
Show worked answer
All contain one 2 and one 3: HCF = 6. Take the greatest required powers for LCM.
Packs with no leftovers
Use 32 red beads and 48 blue beads to make the greatest number of identical packs, using all the beads with none left over. What is in each pack?
Show worked answer
HCF = 16, so make 16 packs. Each has 32 ÷ 16 = 2 red beads and 48 ÷ 16 = 3 blue beads.
Lights flashing together
Lights flash every 15 and 20 seconds, starting together. How long until they next flash together?
Show worked answer
They next flash together after 60 seconds.
No common prime factors
Find the HCF and LCM of 8 and 15.
Show worked answer
There is no shared prime, so HCF = 1.
Recover a number
Two positive integers have HCF 4 and LCM 84. One is 12. Find the other.
Show worked answer
Check: HCF(12, 28) = 4 and LCM(12, 28) = 84.
Examiner-style feedback
Common prime factors, hcf and lcm mistakes
Only shared prime copies belong in the HCF. The result must divide every original number.
2³ is three copies of 2. Your list must preserve all three copies, not just the distinct prime 2.
Ask whether the unknown must divide the totals or be a multiple of each interval. Explain that requirement before choosing HCF or LCM.
30-second recap
Choose, carry out, check
Translate the wording before you reach for a rule.
- Factor: divides exactly. Multiple: a whole-number copy.
- HCF: greatest shared divisor.
- LCM: least positive shared multiple.
- Check the answer against every starting number.
Quick answers
Prime Factors, HCF and LCM FAQ
Is 1 a prime number?
No. It has one positive factor; a prime must have exactly two.
Is the LCM always the product?
No. That works when the numbers have HCF 1. For example, 4 and 6 have LCM 12, not 24.
Content standards
Curriculum and rights review
N4: prime numbers, factors, multiples, HCF, LCM and prime factorisation across tiers. Questions, suggested marks, solutions and diagrams are original practice material, not official past-paper questions.
Official specification references