Factors, multiples and primes — the definitions
These three ideas are easy to mix up, so it helps to be precise:
- Factor — a number that divides exactly into another number, with no remainder. The factors of 12 are 1, 2, 3, 4, 6 and 12.
- Multiple — a number in the times table of another number. The first few multiples of 4 are 4, 8, 12, 16, 20…
- Prime number — a number greater than 1 with exactly two factors: 1 and itself. The first few primes are 2, 3, 5, 7, 11, 13…
Note that 1 is not a prime number (it only has one factor, itself) and 2 is the only even prime number.
Listing factors and multiples
Worked Example 1
List all the factors of 24.
1
Work through pairs systematically, starting from 1: 1×24, 2×12, 3×8, 4×6
2
Check 5 — it doesn't divide exactly into 24, so skip it. The next pair would repeat 4×6, so stop
Answer1, 2, 3, 4, 6, 8, 12, 24
Worked Example 2
List the first 5 multiples of 7.
1
Multiply 7 by 1, 2, 3, 4 and 5 in turn: 7×1, 7×2, 7×3, 7×4, 7×5
Answer7, 14, 21, 28, 35
Prime factorisation
Every whole number greater than 1 can be written as a product of prime numbers — this is called its prime factorisation. A factor tree is the standard way to find it: split the number into any two factors, then keep splitting each factor until every branch ends in a prime number.
Worked Example 3
Write 60 as a product of its prime factors.
1
Split 60 into 6 × 10
2
Split 6 into 2 × 3 (both prime, so these branches stop); split 10 into 2 × 5 (both prime, so these branches stop too)
3
Collect every prime at the end of a branch: 2, 3, 2, 5
4
Write in index form, smallest prime first: 60 = 2² × 3 × 5
Answer60 = 2² × 3 × 5
Factor tree for 60. Reading the circled primes: 60 = 2 × 3 × 2 × 5 = 2² × 3 × 5.
Identifying prime numbers
Worked Example 4
Is 91 a prime number?
1
Test small primes in turn: 91 is odd, so not divisible by 2; digits don't sum to a multiple of 3, so not divisible by 3; doesn't end in 0 or 5, so not divisible by 5
2
Test 7: 91 ÷ 7 = 13 exactly
AnswerNo — 91 = 7 × 13, so it is not prime
Practice questions
Work through each question before checking the answers.
Foundation (Grade 3–5)
Q1List all the factors of 18.Foundation
Q2List the first five multiples of 6.Foundation
Q3Write down all the prime numbers between 1 and 20.Foundation
Q4Is 51 a prime number? Show how you know.Foundation
Q5Write 30 as a product of its prime factors.Foundation
Higher (Grade 5–7)
Q6Write 72 as a product of its prime factors, giving your answer in index form.Higher
Q7Write 108 as a product of its prime factors, giving your answer in index form.Higher
Q8Find the smallest number that is a multiple of both 6 and 8.Higher
Q9Is 143 a prime number? Show how you know.Higher
Q10A number has exactly 3 factors. Explain what this tells you about the number, and give an example.Higher
Higher — Hard (Grade 8–9)
Q11Given that 120 = 2³ × 3 × 5, write down all the factors of 120 that are prime.Grade 8–9
Q12Find the smallest number that has both 90 and 126 as factors, using their prime factorisations.Grade 8–9
Q13A number written as 2ᵃ × 3² has exactly 12 factors. Find the value of a.Grade 8–9
Q14Prove that the sum of two consecutive prime numbers greater than 2 is always even.Grade 8–9
Q15n is a prime number greater than 2. Explain why n² + 1 is always even.Grade 8–9
Answers
Foundation (Q1–Q5)
Q11, 2, 3, 6, 9, 18
Q26, 12, 18, 24, 30
Q32, 3, 5, 7, 11, 13, 17, 19
Q4No — 51 = 3 × 17
Q530 = 2 × 3 × 5
Higher (Q6–Q10)
Q672 = 2³ × 3²
Q7108 = 2² × 3³
Q824(the lowest common multiple of 6 and 8)
Q9No — 143 = 11 × 13
Q10The number must be the square of a prime — e.g. 9 = 3² has factors 1, 3, 9.
Higher — Hard (Q11–Q15)
Q112, 3, 5
Q121890(90 = 2×3²×5, 126 = 2×3²×7; LCM = 2×3²×5×7 = 1890)
Q13a = 3(number of factors = (a+1)(2+1) = 12, so a+1 = 4)
Q14Every prime greater than 2 is odd. Odd + odd = even, so the sum of two consecutive primes greater than 2 is always even.
Q15Since n > 2 and n is prime, n must be odd. An odd number squared is odd, and odd + 1 is even — so n² + 1 is always even.
Common mistakes
Common Mistake 1
Thinking 1 is a prime number
A prime number needs exactly two factors. 1 only has one factor (itself), so it is not prime — the smallest prime number is 2.
Common Mistake 2
Mixing up factors and multiples
Factors of a number are smaller than or equal to it (or equal); multiples are the number's times table, going up forever. "Factors of 12" and "multiples of 12" are completely different lists.
Common Mistake 3
Stopping a factor tree too early
Every branch must end in a prime number. It's easy to stop at a number like 4 or 9, forgetting these can still be split further (4 = 2×2, 9 = 3×3).
Common Mistake 4
Forgetting to write the final answer in index form
When a prime factor appears more than once, GCSE answers are expected in index form — write 2×2×3 as 2² × 3, not as a list of repeated primes.
Exam tips
💡 Exam Tip 1
Test primes in order: 2, 3, 5, 7, 11…
To check if a number is prime, or to start a factor tree, test the small primes in order. If none of them divide in up to the square root of the number, it's prime.
💡 Exam Tip 2
List factors in pairs
Write factors as multiplying pairs (1×24, 2×12, 3×8, 4×6) rather than searching randomly — this makes it far less likely you'll miss one.
💡 Exam Tip 3
Any factor tree for the same number gives the same answer
It doesn't matter which two factors you split first — every valid factor tree for a number ends with the same set of prime factors, just possibly in a different order.
💡 Exam Tip 4
Use prime factorisation for HCF and LCM questions
Writing two numbers as products of primes makes HCF and LCM questions far more reliable than listing factors or multiples by hand, especially for larger numbers.
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