Which of the following is a factor of 12?
A factor divides exactly.
12 ÷ 3 = 4, so 3 is a factor.
Factors and Multiples
Factors divide exactly into a number, while multiples are the result of multiplying a number. These concepts are essential for topics such as HCF, LCM and prime factorisation in GCSE Maths.
Overview
Factors and multiples are key number concepts that appear throughout maths.
They are used when simplifying fractions, applying divisibility rules, finding HCF and LCM, and solving number problems.
Factors of 12: \(1, 2, 3, 4, 6, 12\)
Multiples of 12: \(12, 24, 36, 48, \dots\)
Multiples of 12: \(12, 24, 36, 48, \dots\)
A factor divides exactly into a number, while a multiple is found by multiplying a number. Students often mix these up, so it is important to understand the difference clearly.
What you should understand after this topic
- Understand what factors and multiples are
- List factors and multiples of a number
- Apply factors to divisibility and prime numbers
- Understand how this topic links to HCF and LCM
- Avoid confusing factors and multiples
Key Definitions
Factor
A number that divides exactly into another number.
Multiple
A number found by multiplying a number by an integer.
Common Factor
A factor shared by two or more numbers.
Common Multiple
A multiple shared by two or more numbers.
Highest Common Factor (HCF)
The greatest factor shared by two or more numbers.
Lowest Common Multiple (LCM)
The smallest multiple shared by two or more numbers.
Key Rules
Factors divide exactly
If there is no remainder, it is a factor.
Multiples keep going forever
You can keep multiplying by 1, 2, 3, 4, and so on.
Factors are limited
A number has only a finite set of factors.
Multiples are unlimited
A number has infinitely many multiples.
Quick Comparison
| Number | Some Factors | Some Multiples |
|---|---|---|
| 5 | 1, 5 | 5, 10, 15, 20 |
| 8 | 1, 2, 4, 8 | 8, 16, 24, 32 |
| 12 | 1, 2, 3, 4, 6, 12 | 12, 24, 36, 48 |
How to Solve
What is a factor?
A factor is a number that divides exactly into another number, leaving no remainder.
\( 12 \div 3 = 4 \)
Because 3 divides into 12 exactly, 3 is a factor of 12.
Listing factors
To find all factors, use factor pairs. Work systematically from 1 upwards.
Factor pairs of 18: \(1 \times 18\), \(2 \times 9\), \(3 \times 6\)
So the factors of 18 are: \(1, 2, 3, 6, 9, 18\).
Exam tip: Stop when the pairs start repeating.
What is a multiple?
A multiple is a number you get by multiplying another number by whole numbers.
Multiples of 7: \(7, 14, 21, 28, 35, \dots\)
These come from: \(7 \times 1\), \(7 \times 2\), \(7 \times 3\), etc.
Factors and multiples compared
Factors
Numbers that divide into a number exactly.
Multiples
Numbers found by multiplying a number.
Common factors and HCF
A common factor is shared by two or more numbers. The highest common factor (HCF) is the largest of these.
Factors of 12: \(1, 2, 3, 4, 6, 12\)
Factors of 18: \(1, 2, 3, 6, 9, 18\)
Factors of 18: \(1, 2, 3, 6, 9, 18\)
Common multiples and LCM
A common multiple is shared by two or more numbers. The lowest common multiple (LCM) is the smallest of these.
Multiples of 4: \(4, 8, 12, 16, 20, 24, \dots\)
Multiples of 6: \(6, 12, 18, 24, 30, \dots\)
Multiples of 6: \(6, 12, 18, 24, 30, \dots\)
Common multiples: \(12, 24, 36, \dots\).
LCM = \(12\).
LCM is important in fractions when finding common denominators.
Prime numbers
A prime number has exactly two factors: 1 and itself.
Prime numbers: \(2, 3, 5, 7, 11, 13, \dots\)
For example, 7 has only two factors: 1 and 7, so it is prime.
Useful divisibility rules
Divisible by 2
Ends in 0, 2, 4, 6 or 8
Divisible by 3
Digits add to a multiple of 3
Divisible by 5
Ends in 0 or 5
Divisible by 10
Ends in 0
Example Questions
Edexcel
Exam-style questions inspired by Edexcel GCSE Mathematics.
Edexcel
Write down all the factors of 24.
Edexcel
Write down the first five multiples of 7.
Edexcel
Find the highest common factor (HCF) of \( 18 \) and \( 24 \).
Edexcel
Find the lowest common multiple (LCM) of \( 12 \) and \( 15 \).
Edexcel
Determine whether 91 is a prime number. Give a reason for your answer.
AQA
Exam-style questions based on the AQA GCSE Mathematics specification, focusing on divisibility, prime numbers, and number relationships.
AQA
List all the prime numbers less than 20.
AQA
Write \( 84 \) as a product of its prime factors.
AQA
Find the highest common factor (HCF) of \( 36 \) and \( 54 \).
AQA
Find the lowest common multiple (LCM) of \( 8 \) and \( 20 \).
AQA
A student says that 1 is a prime number.
Tick one box. Yes ☐ No ☐
Give a reason for your answer.
OCR
Exam-style questions aligned with OCR GCSE Mathematics, emphasising reasoning, prime factorisation, and problem-solving.
OCR
Express \( 126 \) as a product of its prime factors.
OCR
Find the highest common factor (HCF) of \( 126 \) and \( 210 \).
OCR
Three bells ring at intervals of 6 minutes, 8 minutes, and 10 minutes. If they ring together at 9:00 am, at what time will they next ring together?
OCR
Find the smallest positive number that is a multiple of both 9 and 12.
OCR
Explain why every multiple of 6 is also a multiple of 3.
Exam Checklist
Step 1
Check whether the question asks for factors or multiples.
Step 2
Use factor pairs when listing factors.
Step 3
For common factors or multiples, compare both lists carefully.
Step 4
Check whether the question wants all common values, HCF or LCM.
Most common exam mistakes
Wrong idea
Listing multiples when the question asked for factors.
Incomplete list
Forgetting one of the factor pairs.
HCF / LCM confusion
Choosing the largest common multiple instead of the lowest.
Prime number errors
Calling a number prime without checking its factors properly.
Common Mistakes
These are common mistakes students make when working with factors and multiples in GCSE Maths.
Mixing up factors and multiples
Incorrect
A student lists multiples when asked for factors.
Correct
Factors divide into a number exactly, while multiples are found by multiplying the number. Make sure you know which one the question is asking for.
Forgetting factors must divide exactly
Incorrect
A student includes numbers that do not divide the original number fully.
Correct
A factor must divide the number with no remainder. If there is a remainder, it is not a factor.
Missing factor pairs
Incorrect
A student lists only small factors and misses larger ones.
Correct
Factors come in pairs. For example, for 12: (1,12), (2,6), (3,4). Always check all pairs to avoid missing any.
Thinking multiples are limited
Incorrect
A student stops listing multiples after a few values.
Correct
Multiples continue indefinitely. For example, multiples of 5 are 5, 10, 15, 20, and so on.
Confusing HCF and LCM
Incorrect
A student finds the highest common factor when asked for the lowest common multiple.
Correct
HCF is the largest number that divides both values, while LCM is the smallest number both values divide into. Make sure you identify the correct one.
Try It Yourself
Practise identifying factors, multiples, primes and common factors.
Foundation
Foundation Practice
Understand factors, multiples and prime numbers.
Write all factors of 10 (ascending, comma separated).
Find numbers that divide 10 exactly.
Factors are 1, 2, 5 and 10.
Which number is a multiple of 6?
Multiples are results of multiplication.
6 × 3 = 18.
Write the first 3 multiples of 4 (comma separated).
Multiply 4 by 1, 2, 3.
4, 8, 12.
Which number is prime?
Prime numbers have exactly 2 factors.
7 has only factors 1 and 7.
Is 11 a prime number? Write yes or no.
Check its factors.
11 has only 1 and 11, so yes.
Which is NOT a factor of 20?
Check division.
20 ÷ 3 is not whole.
Write all multiples of 5 up to 20.
Keep adding 5.
5, 10, 15, 20.
A student says 1 is a prime number. What is wrong?
Count factors of 1.
1 has only one factor, so it is not prime.
How many factors does 9 have?
List them.
Factors are 1, 3, 9 → 3 factors.
Higher
Higher Practice
Find HCF, LCM and use prime factorisation.
Find the HCF of 12 and 18.
Find common factors.
Common factors: 1, 2, 3, 6 → highest is 6.
Find the HCF of 20 and 30.
List factors or use prime factors.
Common highest factor is 10.
Find the LCM of 4 and 6.
Find smallest common multiple.
Multiples: 4 → 4,8,12… and 6 → 6,12… → 12.
Find the LCM of 3 and 5.
Multiply if no common factors.
3 × 5 = 15.
Which is the prime factorisation of 12?
Break into primes.
12 = 2 × 2 × 3.
Write 18 as a product of primes.
Divide by smallest primes.
18 = 2 × 3 × 3.
Find the HCF of 8 and 12 using prime factors.
Find common primes.
8 = 2³, 12 = 2² × 3 → HCF = 2² = 4.
Find the LCM of 8 and 12.
Use highest powers.
LCM = 2³ × 3 = 24.
A student says HCF of 6 and 8 is 48. What is wrong?
HCF is the largest common factor, not multiple.
48 is a multiple, not a factor.
Is 21 a multiple of 7? Write yes or no.
Check division.
21 ÷ 7 = 3, so yes.
Games
Practise this topic with interactive games.
Games coming soon.
Factors and Multiples Video Tutorial
Frequently Asked Questions
What is a factor?
A factor is a number that divides another number exactly.
What is the difference between HCF and LCM?
HCF is the highest common factor, while LCM is the lowest common multiple.
How do I find factors efficiently?
List pairs of numbers that multiply to give the original number.