Prime Factorisation

In this section we shall be looking at prime factorisation. You will be able to understand highest common factors and how to use factor trees to find the highest common factors HCF.

What is a prime number?

Prime numbers are numbers with exactly two factors. These numbers can only be divided by themselves and 1 for example 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, these are all prime numbers. The only numbers which divide into 37 without any remainders in the result are 1 and 37.

What is a factor

A factor is a multiple or a number used in a multiplication to give a certain number, the diagram below shows two factors of a number.

Factor Tree

We can use a factor tree to help us break down any number into a product of its prime factors. Consider for example 80.
First we have to think of two factors which multiply to give 80.
Let’s choose 4×20, you could choose any pair of factors such as 10×8 etc.
Now split 4 and 20 to find their factor pairs 2×2=4 and 4×5=20 The fnumbers you see at the end of each branch cannot be split any further.
The numbers are the prime factors of 80. We can say;

Index Notation

Using prime factors we find that 2x2x2x2x5x5=400. Here we have the number 2 repeating more than once. We call 2 a repeated factor.
We can show repeated factors by using powers or index notation We write 400 as;

Let’s look at the factor tree of 400.

The factor tree shows that;

Using the index notation we get;

Highest common factor (HCF)

The highest common factor (HCF) of 18 and 30 is the highest number that both divide by without having any remainders in the result.

For example

The highest factor that appears in both lists above is a 6. Which means the highest common factor (HCF) of 18 and 30 is a 6.

Finding the highest common factor

We could also use the factor tree to find the highest common factor of two numbers.

Example

Find the HCF of 64 and 120
Here we draw the factor tree for both numbers 64 and 120. Let’s start with 64.

Now we draw the factor tree for 120.

Next we identify which factors occur in both products.

Next we multiply the factors which occur in both list to find the highest common factor.

The highest common factor of 64 and 120 is 8.