List factors and multiples, recognise prime numbers, and use them to solve 'when do they meet again' problems. It is a Foundation topic, aimed at grade 1. Free questions with answers are on this page. Worked examples show each line before you try the questions.
A factor of a number divides into it with no remainder. A multiple is what you get when you times it by 1,2,3,…. 12 is a factor of 36; 36 is a multiple of 12. A prime number has exactly two factors: 1 and itself. 1 is not prime. 2 is the only even prime.
The highest common factor (HCF) is the largest number that is a factor of both. The lowest common multiple (LCM) is the smallest number that is a multiple of both. Listing factors or using prime-factor trees both work; listing is enough at this grade for two modest numbers such as 24 and 36.
To test whether 21,23,27,29 are prime, try dividing by primes up to the square root. 21=3×7 and 27=33 are not prime; 23 and 29 are.
Objectives
List factors in pairs.
List multiples by counting on.
Tell a factor question from a multiple question.
Key points
List all the factors of a number in pairs so none are missed.
List the multiples of a number (its times table).
Decide whether a number is prime.
Find the highest common factor (HCF) and lowest common multiple (LCM) of two small numbers by listing.
Worked examples
Example 1
How many factors does 10 have?
2 marks
Pairs: 1 × 10 and 2 × 5.
The factors are 1, 2, 5 and 10.
There are 4 factors.
Answer: 4
Common slips
Slip: Only 1 and 10. 2 and 5 also multiply to 10.
Slip: The factors are 10, 20, 30. Those are multiples.
Slip: There are 2 factors. Two pairs make four factors.
Example 2
What is the highest common factor of 18 and 24?
2 marks
Factors of 18: 1, 2, 3, 6, 9, 18.
Factors of 24 include 1, 2, 3, 4, 6, 8, 12, 24.
The greatest number in both lists is 6.
Answer: 6
Common slips
Slip: Multiples of 18: 18, 36. A common factor is a divisor, not a multiple.
Slip: 24’s factors stop at 6. 8 and 12 also divide 24.
Slip: The greatest is 72. 72 is a common multiple.
Example 3
What is the lowest common multiple of 6 and 8?
3 marks
Multiples of 8: 8, 16, 24, 32.
24 is also a multiple of 6, because 6 × 4 = 24.
The lowest common multiple is 24.
Answer: 24
Common slips
Slip: Factors of 8: 1, 2, 4, 8. A multiple is at least as big as the number, except 0.
Slip: 16 is a multiple of 6. 16 ÷ 6 is not a whole number.
Slip: The lowest common multiple is 2. 2 is the highest common factor of 6 and 8.
Questions and answers
GCSE Maths factors & multiples questions and answers. Try each one, then open the answer. Easy questions come first.
Question 1 Easy2 marks
What is the highest common factor of 10 and 15?
Show answer
Answer: 5
Shared factors are 1 and 5.
The highest is 5.
M1 — Shared factors are 1 and 5.
A1 — Answer 5
Question 2 Easy2 marks
Is 4 a factor of 22? Answer 1 for yes or 0 for no.
Show answer
Answer: 0
22 ÷ 4 is not a whole number.
The answer is 0.
M1 — 22 ÷ 4 is not a whole number.
A1 — Answer 0
Question 3 Medium2 marks
What is the lowest common multiple of 5 and 6?
Show answer
Answer: 30
5 and 6 share no factor other than 1.
The lowest common multiple is 30.
M1 — 5 and 6 share no factor other than 1.
A1 — Answer 30
Question 4 Medium2 marks
What is the highest common factor of 14 and 21?
Show answer
Answer: 7
Shared factors: 1 and 7.
The highest is 7.
M1 — Shared factors: 1 and 7.
A1 — Answer 7
Question 5 Hard2 marks
What is the lowest common multiple of 9 and 12?
Show answer
Answer: 36
Multiples of 12: 12, 24, 36.
36 ÷ 9 = 4, so the lowest common multiple is 36.
M1 — Multiples of 12: 12, 24, 36.
A1 — Answer 36
Question 6 Hard2 marks
What is the highest common factor of 36 and 48?
Show answer
Answer: 12
Both divide by 12.
The highest common factor is 12.
M1 — Both divide by 12.
A1 — Answer 12
Question 7 Exam-style3 marks
Find the highest common factor of 28 and 42.
Show answer
Answer: 14
28 = 2 × 14 and 42 = 3 × 14, so 14 divides both.
The highest common factor is 14.
M1 — Finds a common factor
M1 — Checks a larger candidate
A1 — Answer 14
Common mistakes
Most lost marks on this topic are one slip.
Mixing up factors and multiples. 12 is a multiple of 4, and 4 is a factor of 12.
Factors divide the number. Multiples are the number times 1, 2, 3…
Forgetting 1 and the number itself when listing factors.
Start at 1 × the number and walk inwards.
Multiplying two numbers for the lowest common multiple when they already share a factor. 4 and 6 give 12, not 24.
If they share a factor, the lowest common multiple is smaller than the product.
Spot the slip
Which list shows all the factors of 6?
1, 2, 3, 6
2, 4, 6
6, 12, 18
Show the slip
1, 2, 3, 6 Factors come in pairs and include 1 and the number 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.
Write factors in pairs on the same line; it makes it obvious when you have finished and stops you missing one.
In an LCM problem, write out both lists of multiples in full — the lists are the method mark, the circled match is the answer mark.
Printable worksheet
The same questions are in a PDF you can print. Answers start on a later page.