What is the difference between the least common multiple and the highest common factor?

Let a and b be two integers.

The lowest common multiple of a and b would be the smallest integer, which is a multiple of a and a multiple of b.

The highest common factor is the greatest integer for which a and b are both multiples, hence note the greatest common factor cannot be greater than the difference between a and b; and therefore if a and b are multiples of the difference between a and b, then the difference between a and b is the highest common factor.

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.

On the other hand, a multiple A of some integer B, exists if we can find some integer C so that C*B=A (A is a multiplication of B). The lowest common multiple (LCM) of two numbers is therefore the lowest number that is multiple 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 and LCM 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. Then to find the LCM of 12 and 8, we use the remaining prime factors that haven't been used yet and multiply the HCF by these. So in this case we have 3 remaining from 12, and 2 remaining from 8, so the LCM is 4*3*2=24.