Definitions:

Highest Common Factor (HCF) – The number with the highest value that can be divided by another number without any remainders.

Lowest Common Multiple (LCM) – The number with the lowest value that two numbers can both be divided into with no remainders.

When finding the Lowest common multiple, the numbers in question must be broken down into their prime factors.
For example: 12 = 2 x 6 = 2 x 2 x 3
Both 2 and 3 are prime numbers so 2, 2, 3 are the prime factors that combine to make up 12.

Once all the numbers in question have been broken down like this, all the prime numbers that are common to all of the numbers are taken in their highest power. By this I mean that, for instance in the above example the number 2 occurs twice and so would be taken to the power of 2, or 2 squared. If a prime number occurs 3 times then it is taken to the power of 3, and so on.

Example 1. Find the LCM of 12, 16 and 8
12 = 2 x 2 x 3
16 = 2 x 8 = 2 x 2 x 4 = 2 x 2 x 2 x 2
8 = 2 x 2 x 2

The prime numbers are then taken in their highest powers and multiplied together:

2 x 2 x 2 x 2 x 3 = 48

To find the highest common factor, the same beginning process is used, so the numbers in question are broken down into all their prime factors. This time however, the answer is the prime number that is common to all in its lowest available form.

So in the example above the HCF would be 2 x 2 which is 4. I will run through another example:

Example 2. Find the HCF of 24, 12 and 30
24 = 2 x 12 = 2 x 2 x 6 = 2 x 2 x 2 x 3
12 = 2 x 6 = 2 x 2 x 3
30 = 2 x 15 = 2 x 3 x 5

The prime factors which occur in all of these are 2 and 3, multiplied together which leaves the answer as 6 which is correct!