LCM from prime factorisation

Listing multiples gets tedious for large numbers. Prime factorisation simplifies the LCM too, and the idea starts with a comparison.

Compare a number with one of its multiples. Take 36 = 2 × 2 × 3 × 3 and 648 = 36 × 18 = (2 × 2 × 3 × 3) × (2 × 3 × 3). What do you notice? The multiple's prime factors CONTAIN the number's prime factors, plus some more. That happens with every multiple.

So if a number is to be a common multiple of two numbers, its factorisation must contain each of their factorisations as a subpart.

Find the LCM of 14 and 35. We have 14 = 2 × 7 and 35 = 5 × 7. A common multiple must contain 2 × 7 as a subpart and 5 × 7 as a subpart. For instance 2 × 7 × 5 × 7 × 3 is a common multiple, and so is 2 × 2 × 5 × 7 × 7 × 11 — but neither is the smallest.

The lowest is 2 × 5 × 7 = 70. It contains 14 = 2 × 7 and it contains 35 = 5 × 7, and removing any factor from it would break one of those. Notice the single 7 serves both.

Now a bigger case: 96 = 2 × 2 × 2 × 2 × 2 × 3 and 360 = 2 × 2 × 2 × 3 × 3 × 5. Build the LCM prime by prime. How many 2s? 96 has five, 360 has three; five 2s cover both, so take five. More than five would still be a common multiple but not the lowest.

How many 3s? 96 has one, 360 has two; take two. How many 5s? 96 has none, 360 has one; take one. So the LCM is 2 × 2 × 2 × 2 × 2 × 3 × 3 × 5 = 1440. The rule: take every prime appearing in EITHER factorisation, each the MAXIMUM number of times it occurs — the mirror image of the HCF rule.