Factors, multiples and primes

• A factor divides exactly into a number. The factors of 12 are 1, 2, 3, 4, 6, 12. Find them in pairs: 1 × 12, 2 × 6, 3 × 4.
• A multiple is in a number's times table: 12, 24, 36, ...
• A prime has exactly two factors, 1 and itself: 2, 3, 5, 7, 11, 13, ... 1 is not prime (it has one factor), and 2 is the only even prime.

Keep dividing by primes, smallest first, until you reach 1.

1. 360 ÷ 2 = 180
2. 180 ÷ 2 = 90
3. 90 ÷ 2 = 45
4. 45 ÷ 3 = 15
5. 15 ÷ 3 = 5
6. 5 ÷ 5 = 1
7. 360 = 2 × 2 × 2 × 3 × 3 × 5 = 2³ × 3² × 5

A factor tree gives the same primes; every number has exactly one prime factorisation.

The highest common factor is the biggest number that divides both.

The lowest common multiple is the smallest number both divide into.

From prime factors: 60 = 2² × 3 × 5 and 72 = 2³ × 3².

• HCF: the primes in both, with the lower power: 2² × 3 = 12.
• LCM: every prime, with the higher power: 2³ × 3² × 5 = 360.

Problems about things that repeat (buses, lights, packs that must match) are LCM problems. Problems about splitting into the biggest equal groups are HCF problems.