How do you find the highest common factor?

Once you have written both numbers as a product of prime factors, to find the highest common factor you need to:

Take the lowest power of numbers that appear in both products of prime factors, and multiply these together

The highest common factor is found by finding all common factors of two numbers and selecting the largest one. For example, 8 and 12 have common factors of 1, 2 and 4.

A longer answer!

To define: a factor B of an integer A, exists if we can find some other integer C so B*C=A (effectively B is a factor of A if it multiples by some other integer to make A, for example 6 has factors 1,6,2 and 3). The highest common factor (HCF) of two numbers is therefore the largest factor that is a factor of both numbers.

We can find the highest common factor by using a factor tree (or called a prime tree), so we start with the number at the top of the tree and then draw two branches off below, using two factors of that number e.g 12 could go into 2 and 6. Then of those new numbers, we leave the prime numbers alone, which in this case is 2, then we split the other numbers further into factors, so 6 goes into 2 and 3, which are both now prime. So we have 2,2,3 as the prime factors of 12. We then get the other number we want to find the HCM of (compared with 12), for example we could choose 8. To find the HCF of 12 and 8, we also split 8 into a factor tree and get 2,2,2. Then the HCF uses the prime factors that exist in both numbers and multiplies them together: here both 12 and 8 have two 2s, so we know the HCF is 2*2=4.