The highest common factor is the largest number that divides into both. The lowest common multiple is the smallest number that both divide into. One looks downwards, the other looks upwards.
For 12 and 18: the factors give 6 as the HCF, and the multiples give 36 as the LCM. The HCF is never bigger than the smaller number, and the LCM is never smaller than the bigger one.
The prime factor method
The reliable method is to break both numbers into prime factors, then read off the answer in two different ways.
| What to do with the primes | Example: 12 and 18 | |
|---|---|---|
| HCF | Only the primes in BOTH, at the LOWEST power | 12 = 2×2×3, 18 = 2×3×3 → 2 × 3 = 6 |
| LCM | EVERY prime in either, at the HIGHEST power | 2×2 × 3×3 = 36 |
For a number to divide into both, it can only use primes that both actually have, and no more copies than the poorer supplies. For a number that both divide into, it must contain everything each of them needs.
Which one does a word problem want?
| The question sounds like | You need |
|---|---|
| Cutting into equal pieces, the largest possible | HCF |
| Sharing into groups with nothing left over | HCF |
| Two repeating events meeting again | LCM |
| The smallest number that fills something exactly | LCM |
An HCF must be less than or equal to the smaller number. An LCM must be greater than or equal to the larger. So 6 cannot be the LCM of 12 and 18, and 36 cannot be the HCF.
A relationship worth knowing
For any two numbers, HCF × LCM = the two numbers multiplied together. For 12 and 18: 6 × 36 = 216, which is 12 × 18.
For any one prime, the HCF takes the lower power and the LCM takes the higher, so between them they take both powers exactly once, which is what multiplying the two numbers does. The rule works only for two numbers.
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HCF: the largest number that divides into both. It looks DOWNWARDS.
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