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Younger Years

A more efficient method for listing the factors of a number

In this blog I will show you the most efficient way to list factors, and explain why this is a good method.

What is a factor?

The factors of a number are numbers that you can divide the number by and get a whole number result.

For example, 5 is a factor of 10 because 10 ÷ 5 = 2 (which is a whole number).

But 4 is not a factor of 10 because 10 ÷ 4 = 2.5 (which is not a whole number).

Every number is guaranteed to have at least two factors – the number itself and 1.

Example 1
Write down all the factors of 12.

Now, the way I see most students approach this is by listing off factors in a random order as they think of them – I like to call it the “scattergun” approach.

So they might say “3 is a factor, so is 2, so is 12 and so is 1. So the answers are 1, 2, 3 and 12”

However, while the numbers we have obtained are all factors of 12, we have missed two factors out – 6 and 4.

The method I use is much more reliable in getting all factors, and it also has an automatic checking process in it!

We are going to write down the factors in pairs.

Every number has 1 as a factor. So we start there, and then try 2, 3, 4 and keep going up in 1’s.

What number do we multiply 1 by to get 12? 1 × 12 = 12, so it’s 12. When we do it like this, every time we get a factor we automatically get another one – reducing the amount of work we need to do. It’s basically “buy one get one free”!

So 1 and 12 are a pair.

Then we move on to 2.

2 × 6 = 12, so 2 and 6 are a pair.  

Then we move on to 3.

3 × 4 = 12, so 3 and 4 are a pair.

Then we move on to 4. We have already had 4, so we can stop there, otherwise we are just going to repeat ourselves.

So the final list of factors is 1, 12, 2, 6, 3 and 4.

Example 2
Write down all the factors of 20.

Again, start with 1.

1 × 20 = 20, so we have 1 and 20.

Then, 2:

2 × 10 = 20, so we have 2 and 10.

Then, 3:

3 is not a factor of 20 because 20 ÷ 3 is not a whole number. So we move on to the next number.

Then, 4:

4 × 5 = 20, so we have 4 and 5.

Then we need to try 5, but we have already had it before so we can stop here.

So the full list of factors is 1, 20, 2, 10, 4 and 5.

Another benefit of this method is that we have a built-in check here. Let’s say you thought that 3 was a factor of 20. You would incorrectly write this number down, but you would quickly realise that it doesn’t work because when you do 20 ÷ 3 you don’t get a whole number. So 3 does not have another number to pair with. Every factor will have a number to pair with. If it doesn’t, then you know something has gone wrong.

Example 3
Write down all the factors of 28.

Start with 1.

1 × 28 = 28, so we have 1 and 28.

Then, 2:

2 × 14 = 28, so we have 2 and 14.

Then, 3:

3 is not a factor of 28 because 28 ÷ 3 is not a whole number. So we move on.

Then, 4:

4 × 7 = 28, so we have 4 and 7.

Then, 5:

5 is not a factor of 28 because 28 does not end in 5 or 0 (if you remember back to the divisibility rules). So we move on.

Then, 6:

6 is also not a factor of 28 because 28 ÷ 6 is not a whole number. So we move on.

Then we need to try 7, but we have already had 7 before so we can stop here.

The full list of factors is 1, 28, 2, 14, 4 and 7.

The final example will be a typical exam-style highest common factor question.

Example 4
Find the highest common factor of 18 and 30.

What we’ll need to do here is list all the factors of 18, then list all the factors of 30 and find the largest number that appears in both lists. It will be very important that we don’t miss any factors out of our lists, as otherwise we could be unlucky and miss out the highest common factor! So our method is going to be very useful here.

First let’s do 18.

You will notice that 1 and the number itself are always the first factors, so we have 1 and 18 automatically.

Then, 2:

2 × 9 = 18, so we have 2 and 9.

Then, 3:

3 × 6 = 18, so we have 3 and 6.

Then, 4:

18 is not divisible by 4, so we move on.

Then, 5:

Same again. 18 is not divisible by 4, so we move on.

Next we try 6, but we have already had 6, so we can stop here.

Factors of 18: 1, 18, 2, 9, 6 and 3.

Now let’s try 30.

1 and 30 are automatically factors.

Then, 2:

2 × 15 = 30, so we have 2 and 15.

Then, 3:

3 × 10 = 30, so we have 3 and 10.

Then, 4:

4 is not a factor of 30 because 30 ÷ 4 is not a whole number.

Then, 5:

5 × 6 = 30, so we have 5 and 6.

6 is next, but we have already had it so we can stop here.

Factors of 30: 1, 30, 2, 15, 3, 10, 5 and 6.

Now I’ll put these two lists next to each other and underline the numbers that appear in both lists (ie. the common factors).

Factors of 18: 1, 18, 2, 9, 3 and 6.
Factors of 30: 1, 30, 2, 15, 3, 10, 5 and 6.

The largest underlined number is 6, so this is the answer. The highest common factor of 18 and 30 is 6.

So there it is – I strongly recommend using this method when listing factors. It is more reliable and will lead to fewer mistakes. To practice this method further, try out our factors and multiples worksheet.

If you’re in Bristol and you or your son/daughter needs some face-to-face help on factors or any other aspect of mathematics, book in a free taster session.

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