HCF and LCM from Primes

Break two numbers into their prime building blocks, lined up by prime. The blocks they share make the highest common factor. All the blocks between them, taking the taller stack of each prime, make the lowest common multiple.

First number

Second number

HCF — the shared primes

Take the smallest stack of each prime the two numbers have in common. Multiply those together and you have the highest common factor.

LCM — all the primes

Take the tallest stack of every prime that appears in either number. Multiply those together and you have the lowest common multiple.

The product rule

For any two numbers, HCF × LCM = the product of the two numbers. The blocks make it obvious: between them, the HCF and LCM use each prime exactly as many times as the two numbers did together.