Skip to 0 minutes and 7 secondsPAULA KELLY: In the last two steps, we looked at multiples. Is it possible for a number to be a multiple of more than one number? If you look back to the times table grid, the answer is clearly yes. 8, for example, appears four times, in the 1 times table, the 8 times table, the 2, and also the 4 times table. This tells us 8 is a common multiple of 1, 2, 4, and 8. Which times tables does the number 12 appear in? The number 12 appears in the following times tables, 1, 2, 3, 4, 6, and 12. So 12 is a multiple of 1, 2, 3, 4, 6, and 12.

Skip to 0 minutes and 52 secondsSo the number 12 can be made by combining these numbers, so 1 multiplied by 12, 2 multiplied by 6, and 3 multiplied by 4. We call the numbers 1, 2, 3, 4, 6, and 12 the factors of 12. So 12 is a multiple of 1, 2, 3, 4, 6, and 12. So 1, 2, 3, 4, 6, and 12 are the factors of 12. We can see in this diagram how the factors of 12 pair up. So all numbers seem to have an even number of factors, as you could always pair them up. This is a very sensible suggestion and true for most numbers. However, let's have a look at the number 16.

Skip to 1 minute and 40 seconds16 appears in the following times tables, 1, 2, 4, 8, and 16. This means that 16 is a multiple of 1, 2, 4, 8, and 16, but also means that 1, 2, 4, 8, and 16 are factors of 16. Let's try to pair them up. 1 and 16 pair up, as 1 multiplied by 16 gives us 16. 2 and 8 pair up, as 2 multiplied by 8 gives us 16. This leaves 4 without a number to pair it with. This is because 4 multiplied by 4 gives us 16. We didn't include the number 4 twice, so 16 has an odd number of factors.

Skip to 2 minutes and 32 secondsFind all the factors of 36, place them in ascending order, pair them up, and see which factor is the odd one. All square numbers have an odd number of factors. Only square numbers have an odd number of factors. If a number has an odd number of factors, it must be a square number.

Multiples and factors

It is easy to get the definitions of ‘multiples’ and ‘factors’ mixed up. Think about the numbers 4 and 12. How would you complete the following sentences using ‘multiple’ and ‘factor’?

12 is a ______ of 4 because 12 is in the 4 times table.
4 is a ______ of 12 because 4 divides exactly into 12 without leaving a remainder.

In this video Paula explores the link between factors and multiples and shows that in most cases the factors of a number can be paired up. However, something unusual happens when we attempt to pair up the factors of a square number.

Problem worksheet

Now complete question 4 from this week’s worksheet.

Teaching resource

This collection contains resources covering topics including factors and multiples. Each topic is introduced in the form of a puzzle, many of which have several different solutions.

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This video is from the free online course:

Maths Subject Knowledge: Understanding Numbers

National STEM Learning Centre