One of the big challenges when working in maths and numbers is trying to work out how to keep them small before making them big. For example what is 8 multiplied by 16? You may know this without the need for paper or even a calculator as you are practiced in the art of mental arithmetic. If, however, you don’t know your 16 times table then there are other ways of working this out – understanding the factors of numbers.

What is a factor?

Simply a factor is an integer (whole number) that can divided a larger integer to produce another integer or a multiple of a number. As a factor multiples by another factor they are often called factor pairs. The smallest factors of any number will be prime number as a prime number can only be divided by 1 and itself (e.g. 2, 3, 5, 7, … 29…). The factor pair for a prime number is 1 and the number itself. Sounds a bit complicated and what’s the point?

Why do I need to understand the factors of a number?

An excellent question. For the most part you don’t. You can use the use the number you have a work it with other numbers. But, there are two examples where knowing the factors of a number is useful.

1 Simplify arithmetic

How to multiply large numbers without a calculator

To answer our question early (8 x 16) we can break it down into some of factors: 8 is 2 x 4, and 16 is 2 x 8. So we can say 2 x 4 x 2 x 8 or 2 x 8 x 8 which is 2 x 64 (I know my square numbers) which is 2 x 60 + 2 x 4 = 128. This is obviously longer that just doing the maths but it helps to build up a solution using simpler multiplications. This especially true when working with big numbers. Breaking a number down into it’s smallest parts (multiples is called factorising. If you keep factoring the factors you will eventually get to prime number number factors. These are smallest factors of an number. When you compare two numbers they will share factors even if that is just 1. When two numbers share multiple factors we can use the largest/highest/greatest common factor (HCF) to reduce both numbers if we wished (I’ll get to that in a minute) to simplify a calculation. The opposite of the highest common factor is the lowest common multiple (LCM). LCM is handy to know what is the smallest number that two numbers can be divided

Let’s try another example: what is 110 x 110? Using the factors of 110 I can tell 10 and 11 are factor pairs (10 x 11 = 110). Knowing this I can split the multiplication like this

10 x 11 x 10 x 11 => 10 x 10 x 11 x 11 => 100 x 11 x 11 => 100 x 121 => 12100

Great. But I could have done all of that with a calculator! True but being able to do maths in your head or on paper is a) very useful and b) often used to test the mathematical skills of a person for example in job interviews. And then there is the opposite of multiplication – division

How to divide large numbers without a calculator

Breaking numbers down into the factor pairs is even more powerful when it comes to division as we can cancel out common factors to both numbers. Before I get into this a super quick reminder on division.

Reminder on division

We use division in two ways in maths. One to divide or quotient something (divide up) and to simply something to a smaller term (divide down). Example one – I have 10 sweets and 5 children: how many sweets should the children get? 10 divided into 5 bags = 2 sweets each. Example two – what is 10/5 in it’s simplest term? 10/5 = 2/1 = 2.

The reason for the clarification is that it’s the second example that knowing factors is helpful rather than the first (children want to count the sweets going into their bag not worry about the maths).