NumberFoundation + Higher★ Core topic
Factors, Multiples
& Prime Numbers.
A factor divides into a number exactly. A multiple is what you get in the times table. A prime has exactly two factors: 1 and itself. From these three ideas, we unlock HCF, LCM and prime factorisation.
I · Key definitions
Factor
A number that divides exactly into another number (no remainder).
factors of 12: 1, 2, 3, 4, 6, 12
Multiple
A number in the times table of another number.
multiples of 6: 6, 12, 18, 24…
Prime
A number with exactly two factors: 1 and itself.
2, 3, 5, 7, 11, 13, 17…
HCF
Highest Common Factor — biggest factor shared by both numbers.
HCF of 12, 18 = 6
LCM
Lowest Common Multiple — smallest multiple shared by both.
LCM of 4, 6 = 12
Square · cube
Square = n × n. Cube = n × n × n.
5² = 25 · 3³ = 27
II · The first few primes
2
3
5
7
11
13
17
19
23
29
31
37
41
43
47
1 is NOT prime (only one factor). 2 is the only even prime — every other even number has 2 as a factor. Memorise these 15 — they cover most GCSE questions.
III · Prime factor tree
72 = 2 × 2 × 2 × 3 × 3 = 2³ × 3²
Keep branching until every leaf is prime. Collect them in index form.
IV · Rules & formulas
Finding HCF (prime factorisation)
1. List primes of each
2. Take shared primes (lowest power)
3. Multiply
72 = 2³×3² · 108 = 2²×3³ → HCF = 2²×3² = 36
Finding LCM (prime factorisation)
1. List primes of each
2. Take all primes (highest power)
3. Multiply
72 = 2³×3² · 108 = 2²×3³ → LCM = 2³×3³ = 216
Handy identity
HCF(a,b) × LCM(a,b) = a × b
HCF(72,108) × LCM(72,108) = 36 × 216 = 7776 = 72 × 108 ✓
V · Divisibility tests
÷ 2
last digit is even
138 ✓ · 137 ✗
÷ 3
digit sum ÷ 3
126: 1+2+6=9 ✓
÷ 4
last two digits ÷ 4
316 (16÷4=4) ✓
÷ 9
digit sum ÷ 9
243: 2+4+3=9 ✓
VI · Worked examples
Find the HCF and LCM of 72 and 108 using prime factorisation.
i.Prime factorise both: 72 = 2 × 2 × 2 × 3 × 3 = 2³ × 3²
ii.108 = 2 × 2 × 3 × 3 × 3 = 2² × 3³
iii.HCF: shared primes, lowest power → 2² × 3² = 4 × 9 = 36
iv.LCM: all primes, highest power → 2³ × 3³ = 8 × 27 = 216
HCF = 36 · LCM = 216
Prime factorise a number
2 marks
Express 360 as a product of prime factors, in index form.
i.360 ÷ 2 = 180 → 180 ÷ 2 = 90 → 90 ÷ 2 = 45
ii.45 ÷ 3 = 15 → 15 ÷ 3 = 5 → 5 is prime, stop
iii.Collect: three 2's, two 3's, one 5
360 = 2³ × 3² × 5
VII · Common mistakes & examiner tips
Common mistakes
Saying 1 is prime. It has only one factor — not two.
Forgetting 2 is prime. It's the only even prime.
Confusing factor and multiple. Factor divides into; multiple is what you get from times tables.
HCF vs LCM muddle. HCF is shared & lowest; LCM is all & highest.
Stopping the factor tree early. Keep going until every leaf is prime.
Examiner tips
Always write in index form (2³×3²) — neater & expected by examiners.
Draw Venn diagrams of primes if struggling with HCF/LCM.
Start with smallest primes (2, 3, 5, 7) when factorising.
Check: HCF × LCM should equal the product of the two numbers.