Factors, Multiples & Prime Factorisation

What factors and multiples really are (and how never to mix them up), what makes a number prime, and how to break any number into its prime factors with a factor tree — worked examples, a clear diagram and an exam-style worksheet with memo.

Factors, multiples and prime numbers are the vocabulary of number work. Get these three words rock-solid now, because the next lesson — HCF and LCM — is built entirely on them, and so is simplifying fractions later this term.

1Factors

A factor of a number divides into it exactly, leaving no remainder.

The trick to listing every factor without missing any is to hunt in pairs, starting from 1 and working up until the pair meets in the middle.

Worked ExampleList all the factors of 24

1
Pairs: 1 × 24, 2 × 12, 3 × 8, 4 × 6.
Work upward; stop when the two numbers in a pair get close together.
2
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24.
Read both numbers from each pair, in order — eight factors.

2Multiples

A multiple of a number is that number times 1, 2, 3, 4, …

Worked ExampleWrite down the first five multiples of 6

1
6 × 1, 6 × 2, 6 × 3, 6 × 4, 6 × 5.
Just keep counting up in that number.
2
6, 12, 18, 24, 30.
Unlike a factor list, a multiple list never ends.

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Memory hook: factors are few (a short, finite list) and multiples are many (they run on forever).

3Prime numbers

A prime number has exactly two factors — 1 and itself: 2, 3, 5, 7, 11, 13, 17, 19, … Two things examiners love to test: 1 is not prime (it has only one factor), and 2 is the only even prime.

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Watch out: learners often call 1 (or even 9, 15, 21) prime. 1 is not prime; and 9 = 3×3, 15 = 3×5, 21 = 3×7 are all composite. The smallest prime is 2.

4Prime factorisation with a factor tree

Every whole number bigger than 1 can be written as a product of primes in exactly one way. The neatest method is a factor tree: split off the smallest prime you can each time, until every branch ends in a prime.

Factor tree for 48 — split off a prime at each step until you reach a prime, then collect them: 48 = 2⁴ × 3.

Worked ExampleWrite 60 as a product of prime factors

1
60 ÷ 2 = 30, 30 ÷ 2 = 15, 15 ÷ 3 = 5, and 5 is prime.
Keep dividing by the smallest prime that fits.
2
60 = 2 × 2 × 3 × 5.
Collect every prime you divided by.
3
60 = 2² × 3 × 5.
Use powers for repeated primes — the exam-ready form.

Practice exercises

Work each one out, then click to reveal the answer.

📄 Worksheet ✅ Memo 📝 Exam paper 🗝️ Marking guide

1
List all the factors of 36.
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1, 2, 3, 4, 6, 9, 12, 18, 36
2
List all the factors of 40.
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1, 2, 4, 5, 8, 10, 20, 40
3
Write down the first five multiples of 8.
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8, 16, 24, 32, 40
4
Is 51 a prime number? Give a reason.
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No — 51 = 3 × 17, so it has factors besides 1 and itself.
5
Write down all the prime numbers between 20 and 30.
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23 and 29
6
Write 90 as a product of its prime factors.
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90 = 2 × 3² × 5 (= 2 × 3 × 3 × 5)
7
Write 84 as a product of its prime factors.
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84 = 2² × 3 × 7 (= 2 × 2 × 3 × 7)