Finding the LCM or HCF from the prime factorisation.

This is a general approach which always works, for any number of numbers.

First, find the full prime factorisations of each number. Try to write them lined up so that the same factor is lined up in each number.

Now look through every prime factor which occurs in any of the numbers. For the LCM, choose the highest index for that factor which appears in any number. For the HCF, choose the lowest. If it doesn't appear in a number then the index is 0, and this factor wont be in the HCF.

For example, take 8, 10, and 24.

8 = 2³

10 = 2 x 5 = 2¹ x 5¹

24 = 2³ x 3¹

Looking at all the factors which appear, 2, 3 and 5, the lowest index of 2 is 1 (in 10). The highest is 3.

The lowest index of 3 is 0 - it doesn't appear in 8 or 10. The highest index is 1, in 24.

The lowest index of 5 is 0 - it doesn't appear in 8 or 24. The highest index is 1, in 10.

So the HCF is 2¹ x 3⁰ x 5⁰ = 2

The LCM is 2³ x 3¹ x 5¹ = 120