Factors, multiples, HCF & LCM

A factor of a number divides into it with no remainder. The factors of 12 are 1, 2, 3, 4, 6 and 12. A multiple is the result of multiplying: the multiples of 4 are 4, 8, 12, 16, … A prime number has exactly two factors — 1 and itself — like 2, 3, 5 and 7.

Every whole number above 1 can be written as a product of primes. Keep dividing by the smallest prime that fits: 12 = 2 × 6 = 2 × 2 × 3, and 90 = 2 × 45 = 2 × 3 × 3 × 5. These prime building blocks are the key to HCF and LCM.

The HCF of two numbers is the biggest number that divides both. Using primes, 12 = 2 × 2 × 3 and 18 = 2 × 3 × 3; the shared primes are 2 × 3 = 6, so the HCF of 12 and 18 is 6. The HCF answers questions like the largest equal group you can make.

The LCM of two numbers is the smallest number both divide into. From the primes of each, take the highest power of every prime that appears: 6 = 2 × 3 and 8 = 2 × 2 × 2 give an LCM of 2 × 2 × 2 × 3 = 24. The LCM answers “when do two repeating events line up again?”