Finding Common Ground

Prime factorisation, highest common factor (HCF) and lowest common multiple (LCM), and the problems each one solves.

Key ideas

Every number above 1 is a product of primes in only one way, e.g. 600 = 2³ × 3 × 5².
HCF: multiply the lowest power of each prime that is common to all the numbers.
LCM: multiply the highest power of every prime that appears in any of the numbers.
Co-prime numbers have HCF 1, so their LCM is their product.
HCF answers “largest equal piece or tile”; LCM answers “when do they next meet or ring together”.

Two numbers	HCF × LCM = product of the numbers
Example	675 = 3³ × 5², 600 = 2³ × 3 × 5²: HCF = 3 × 5² = 75, LCM = 2³ × 3³ × 5² = 5400
Co-prime	HCF = 1, LCM = a × b

Write 675 and 600 in prime-power form, then find their HCF and LCM.

Working: HCF: lowest power of each common prime; LCM: highest power of every prime

Answer: 3³ × 5², 2³ × 3 × 5²; HCF 75, LCM 5400