Prime Numbers, Factors and Multiples Explained for KS3 Maths

A factor of a number divides into it exactly with no remainder. A multiple of a number is what you get when you multiply it by a positive integer. A prime number has exactly two factors: 1 and itself. These three ideas underpin nearly every number topic in KS3 and GCSE maths.

What is a factor?

A factor of a whole number n is any integer that divides n exactly (with zero remainder).

Example: The factors of 12 are 1, 2, 3, 4, 6, and 12 — because each divides 12 exactly.

12 ÷ 1 = 12, 12 ÷ 2 = 6, 12 ÷ 3 = 4, 12 ÷ 4 = 3, 12 ÷ 6 = 2, 12 ÷ 12 = 1.

Notice that the factors come in pairs that multiply to 12: (1, 12), (2, 6), (3, 4).

How to list all factors of a number

Work through factor pairs systematically, starting from 1:

Find all factors of 30:

Pair | Check
1 × 30 | ✓
2 × 15 | ✓ (30 ÷ 2 = 15)
3 × 10 | ✓ (30 ÷ 3 = 10)
4 × ? | 30 ÷ 4 = 7.5 — not a whole number ✗
5 × 6 | ✓ (30 ÷ 5 = 6)
6 × 5 | already found — stop

Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30.

Stop when the factor being tested is greater than the square root of the number (√30 ≈ 5.5), as any factor beyond that will already have appeared in a pair.

What is a multiple?

A multiple of a number is produced by multiplying that number by a positive integer (1, 2, 3, 4, …).

Multiples of 7: 7, 14, 21, 28, 35, 42, 49, 56, …

The list of multiples is infinite.

Tip: To check if a number is a multiple of another, see if the division leaves zero remainder. Is 84 a multiple of 7? 84 ÷ 7 = 12 exactly — yes.

What is a prime number?

A prime number has exactly two factors: 1 and itself.

2 is prime: factors are 1 and 2. (2 is the only even prime.)

3 is prime: factors are 1 and 3.

5 is prime: factors are 1 and 5.

7 is prime: factors are 1 and 7.

1 is NOT prime. It has only one factor (itself). By definition, prime numbers have exactly two factors; 1 has exactly one.

Primes up to 30: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. Memorise these.

How to test if a number is prime

Divide by all primes up to its square root. If none divide it exactly, it is prime.

Is 97 prime? √97 ≈ 9.8. Test primes up to 9: 2, 3, 5, 7.

97 ÷ 2 = 48.5 ✗

97 ÷ 3: digit sum 9 + 7 = 16, not divisible by 3 ✗

97 ÷ 5: doesn't end in 0 or 5 ✗

97 ÷ 7 = 13.857… ✗

97 is prime.

Prime factor decomposition

Every integer greater than 1 can be written as a product of prime numbers in exactly one way (the Fundamental Theorem of Arithmetic). This is called prime factor decomposition or prime factorisation.

Using a factor tree

Example: express 60 as a product of its prime factors.

60
/ \
6 10
/ \ / \
2 3 2 5

Primes: 2, 3, 2, 5.

60 = 2 × 3 × 2 × 5 = 2² × 3 × 5

(Conventionally, repeated primes are written as powers and the factors are listed in ascending order.)

Example: express 84 as a product of its prime factors.

84
/ \
4 21
/ \ / \
2 2 3 7

84 = 2² × 3 × 7

Verify

2² × 3 × 7 = 4 × 3 × 7 = 12 × 7 = 84 ✓

Highest Common Factor (HCF)

The HCF of two numbers is the largest number that is a factor of both.

Method using prime factor decomposition:

Write each number as a product of prime factors.

Identify the prime factors common to both.

Multiply the common factors together (using the lower power where both share the same prime).

Worked example — HCF of 36 and 60

36 = 2² × 3²

60 = 2² × 3 × 5

Common primes: 2 (lower power is 2²) and 3 (lower power is 3¹).

HCF = 2² × 3 = 4 × 3 = 12

HCF(36, 60) = 12

Check: 36 ÷ 12 = 3 ✓; 60 ÷ 12 = 5 ✓.

Lowest Common Multiple (LCM)

The LCM of two numbers is the smallest number that is a multiple of both.

Method using prime factor decomposition:

Write each number as a product of prime factors.

Take each prime factor to its highest power that appears in either number.

Multiply these together.

Worked example — LCM of 12 and 18

12 = 2² × 3

18 = 2 × 3²

Primes present: 2 (highest power: 2²) and 3 (highest power: 3²).

LCM = 2² × 3² = 4 × 9 = 36

LCM(12, 18) = 36

Check: 36 ÷ 12 = 3 ✓; 36 ÷ 18 = 2 ✓.

HCF vs LCM — which is which?

HCF — the largest number that divides both (useful for simplifying fractions, sharing things out equally).

LCM — the smallest number that both divide into (useful for finding common denominators).

A useful check: HCF × LCM = product of the two original numbers. For 12 and 18: HCF × LCM = 6 × 36 = 216 = 12 × 18 ✓.

How prime numbers and factors fit the KS3 national curriculum

The Department for Education's KS3 mathematics programme of study requires pupils to "use the concepts and vocabulary of prime numbers, factors (or divisors), multiples, common factors, common multiples, highest common factor, lowest common multiple, prime factorisation, including using product notation." BBC Bitesize's KS3 number resources confirm that HCF and LCM appear in both calculator and non-calculator assessments throughout Years 7, 8, and 9, and that fluency with factor trees is a prerequisite for simplifying algebraic fractions and solving problems involving ratio and proportion at GCSE.

Common mistakes

Mistake 1 — Including 1 as a prime number. 1 is not prime. It has only one factor. The question "write 12 as a product of prime factors" should give 2² × 3, not 1 × 2² × 3.

Mistake 2 — Missing factor pairs. When listing factors, work systematically in pairs from 1 upwards. A common error is to miss 4 as a factor of 20 (since 1 × 20, 2 × 10, 4 × 5 are the full set, and students sometimes stop after 1, 2, 5, 10, 20).

Mistake 3 — Confusing HCF and LCM. HCF is found by taking the lowest powers of shared primes (the overlap). LCM is found by taking the highest powers of all primes (the union). If you mix them up, the HCF answer will be larger than one or both original numbers — an impossible result.

Mistake 4 — Stopping a factor tree too early. If a branch gives 6, it must be split further into 2 × 3. Every branch must end in a circled prime.

Frequently asked questions

Is 2 the only even prime number?

Yes. Every even number greater than 2 is divisible by 2 (and therefore has more than two factors), so it cannot be prime. 2 is the only even prime because it has exactly two factors: 1 and 2.

What is the difference between a factor and a multiple?

A factor of n divides n exactly and is always less than or equal to n. A multiple of n is produced by multiplying n by a positive integer and is always greater than or equal to n. For example, 4 is a factor of 12, and 12 is a multiple of 4. The relationship is symmetric: if a is a factor of b, then b is a multiple of a.

Why do we need prime factor decomposition?

Prime factorisation provides a unique "fingerprint" for every integer. This makes it straightforward to calculate HCF and LCM, simplify fractions (divide numerator and denominator by their HCF), and check divisibility. It also underpins modern encryption — the difficulty of factorising very large numbers into primes is the basis of RSA cryptography, which secures internet banking.

How many prime numbers are there?

Infinitely many. In 300 BCE, Euclid proved that if you assume you have found all the primes and multiply them together, adding 1 gives a number not divisible by any of them — so there must be at least one more prime not in your list. This argument shows the list never ends.

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