HCF and LCM from primes

First, what the two words mean:

The prime method, step by step. Write each number as powers of primes. If a prime is missing from a number, you can think of it as that prime to the power 0:

6 = 2¹ × 3¹ × 5⁰ and 20 = 2² × 3⁰ × 5¹

• HCF: take each common prime to its smallest power: 2¹ = 2.
• LCM: take every prime to its greatest power: 2² × 3 × 5 = 60.

It works for three numbers too. 6 = 2 × 3, 72 = 2³ × 3², 120 = 2³ × 3 × 5, so HCF = 2 × 3 = 6 and LCM = 2³ × 3² × 5 = 360.

A shortcut for two numbers only: HCF × LCM = the product of the two numbers. Check: 2 × 60 = 120 = 6 × 20. So if you know the HCF, the LCM is quick: LCM(96, 404) = 96 × 404 ÷ 4 = 9696.

Careful: this shortcut fails for three numbers. 6 × 72 × 120 = 51840, but HCF × LCM = 6 × 360 = 2160.

A quick check: the HCF always divides the LCM. If someone claims HCF 7 and LCM 60, something is wrong, because 7 does not divide 60.

Word problems: HCF or LCM? Don't hunt for keywords. Ask: will the answer be bigger or smaller than the numbers given?

• Sonia goes round a field in 18 minutes, Ravi in 12. When are they both back at the start? Could it be after 6 minutes? No, neither has finished a round. The answer must be bigger, so use the LCM: 36 minutes.
• Cutting things into the biggest equal pieces, or putting the most people in each room, gives an answer smaller than the numbers, so use the HCF.