Factors, Multiples and Primes (HCF & LCM)

GCSE Maths · Number

Key definitions

Factor – divides exactly into a number. Factors of 12: 1, 2, 3, 4, 6, 12.
Multiple – the times-table of a number. Multiples of 4: 4, 8, 12, 16…
Prime – exactly two factors (1 and itself): 2, 3, 5, 7, 11, 13… 1 is not prime; 2 is the only even prime.

Prime factor decomposition

Break a number into a product of primes with a factor tree:

60 = 2 × 30 = 2 × 2 × 15 = 2 × 2 × 3 × 5
   = 2² × 3 × 5

HCF — Highest Common Factor

Multiply the prime factors that appear in both numbers.

12 = 2² × 3
18 = 2  × 3²
HCF = 2 × 3 = 6

LCM — Lowest Common Multiple

Multiply each prime factor the highest number of times it appears in either number.

LCM = 2² × 3² = 36

Quick relationship

HCF × LCM = product of the two numbers → 6 × 36 = 12 × 18 = 216 ✓.

Venn diagram method

Put shared primes in the overlap (multiply = HCF); multiply everything in the diagram = LCM.

Exam tip

HCF asks for the biggest number that divides both; LCM asks for the smallest number both go into. Word problems with "every X minutes / every Y minutes, when do they coincide?" are LCM.