**Prime Factors, HCF and LCM**

Understanding prime factors is important to be able to find the **Lowest Common Multiple (LCM)** and **Highest Common Factor (HCF)** of two or more numbers. Make sure you are happy with the following topics before continuing.

**Prime Factorisation and Factor Trees**

We define a **prime factor** of any given number to be any factor that the number has, that is also a **prime number**.

Every positive whole number has a unique prime factorisation – a list of prime numbers that, when multiplied together, give you the original number. In more complicated cases, we use something called a **factor tree**.

**Example: **Determine the prime factorisation of 60.

**Step 1:** To construct a **factor tree**, think of 2 numbers which multiply together to make 60 – here, we’ve gone with 10 and 6.

**Step 2:** Draw two branches coming down from 60, and at the end of the branches write the two factors that you chose.

**Step 3:** If a **factor is prime**, then circle it. If a factor is not prime, then repeat the process as shown in the **factor tree** below.

**Step 4: **The prime factorisation of 60 is therefore

60 = 2 \times 2 \times 3 \times 5

**Step 5: **We write this prime factorisation in **index form**, where if there is more than one of the same factor, we write it as a power instead, where the power is the number of times it occurs. So

60= 2^2 \times 3\times 5

**Highest Common Factor – HCF**

The **Highest Common Factor**, or **HCF**, of two numbers is the biggest number that goes into both of them.

**Example:** Consider the numbers 12 and 20

The factors of 12 are: 1, 2, 3, 4, 6, and 12

The factors of 20 are: 1, 2, 4, 5, 10, and 20

They have a few factors in common, but the **biggest factor they have in common** is 4, therefore 4 is the **HCF** of 12 and 20.

**Lowest Common Multiple – LCM**

The **lowest common multiple**, or** LCM**, of two numbers is the smallest number that is a multiple of both of them.

**Example:** Consider the numbers 5 and 7

Multiples of 5 are: 5, 10, 15, 20, 25, 30, 35, 40, 45, …

Multiples of 7 are: 7, 14, 21, 28, 35, 42, … and so on.

So, we can see that the first occurrence of a number which is a multiple of both of these numbers is 35, therefore 35 is the **LCM **of 5 and 7.

**HCF and LCM – Venn Diagrams**

For large numbers, the easiest way to find the **HCF** and **LCM** is to use Venn diagrams.

**Example:** Find the HCF and LCM of 60 and 27.

**Step 1: **We first need the prime factorisation of both numbers, in which we would use factor trees. However we already have the prime factorisation of 60, which is

60 = 2 \times 2 \times 3 \times 5 = 2^2 \times 3 \times 5 and 27= 3 \times 3 \times 3 = 3^3

**Step 2: **Now, we draw a Venn diagram where one circle is for prime factors of 60 and one circle is for prime factors of 27.

**Step 3:** Looking at the list of factors, if one is shared by both numbers, then we will put it in the intersection and cross it off both lists.

\textcolor{red}{60} = 2 \times 2 \times \bcancel{3} \times 5 and \textcolor{blue}{ 27} = \bcancel{3} \times 3 \times 3

**Step 4:** Any factors that are not shared and haven’t been crossed out, we put in their respective circles.

**Step 5: **To find the HCF, we multiply the numbers in the intersection (these are the factors that are common between both numbers). Here there is only one number, so

**HCF** = 3

**Step 6: **To find the LCM, we multiply all of the numbers in the Venn diagram together. So

**LCM** = 2 \times 2 \times 5 \times 3 \times 3 \times 3

**Example: Prime Factor Tree**

Find the LCM and HCF of 420 and 132.

**[4 marks]**

To do this method, we require the full prime factorisation of both 420 and 132. So, we’re going to use the factor tree method.

The prime factor tree for 420 can be seen on the right,

This gives,

2\times2\times3\times 5\times 7 = 2^2 \times 3 \times 5 \times 7

Going through the same process, we get that the full prime factorisation of 132 is

2\times2\times 3\times 11 = 2^2 \times 3 \times 11

So, now that we have the prime factorisation, we need to draw a Venn diagram where one circle is for prime factors of 420 and one circle is for prime factors of 132.

Looking at the lists of factors, if one is shared by both numbers, then we will put it in the intersection and cross it off both lists.

Then, any factors that aren’t shared, and so haven’t been crossed out, will be put in their respective circles.

To find the HCF is to multiply the numbers in the intersection:

HCF =2\times2\times3=12

To find the LCM, all we need to do is multiply all the numbers now in the Venn diagram together:

LCM =5\times7\times2\times 2\times3\times11=4620