Prime Factors LCM HCF Worksheets, Questions and Revision

Prime Factors LCM HCF Worksheets, Questions and Revision

GCSE 4 - 5KS3AQAEdexcelOCRWJECFoundationAQA 2022Edexcel 2022OCR 2022WJEC 2022

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.

Level 4-5 GCSE KS3 AQA Edexcel OCR WJEC

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.

Prime Factor Tree
Prime Factor Tree

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

Level 4-5GCSEKS3AQAEdexcelOCRWJEC

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.

Level 4-5 GCSE KS3 AQA Edexcel OCR WJEC

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.

Level 4-5 GCSE KS3 AQA Edexcel OCR WJEC
Level 4-5 GCSE KS3 AQA Edexcel OCR WJEC

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

HCF LCM Venn Diagram
HCF LCM Venn Diagram

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

Level 4-5GCSEKS3AQAEdexcelOCRWJEC
Level 4-5 GCSE KS3 AQA Edexcel OCR WJEC

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

Prime Factor Tree (2)

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.

HCF LCM Venn Diagram
HCF LCM Venn Diagram

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

Level 4-5GCSEKS3AQAEdexcelOCRWJEC

Example Questions

The prime factors of a number can be displayed using a prime factor tree.

The prime factorisation of 72 is,

 

72=2\times2\times2\times3\times3

Written in index notation, the answer is,

72=2^3\times3^2

The prime factors of a number can be displayed using a prime factor tree.

The prime factorisation of 140 is,

 

 

140=2\times2\times5\times7

 

Written in index notation, the answer is,

140=2^2\times5\times7

First, we have to find the prime factorisation of 24 and of 40:

 

Prime factors of 242\times2\times2\times3

Prime factors of 40: 2\times2\times2\times5

 

To find the HCF, find any prime factors that are in common between both numbers.

 

HCF2\times2\times2=8

 

Next, cross any numbers used so far off from the products.

Prime factors of 24\cancel{2}\times\cancel{2}\times\cancel{2}\times3

Prime factors of 40: \cancel{2}\times\cancel{2}\times\cancel{2}\times5

 

To find the LCM, multiply the HCF by all the factors that have not been crossed out so far.

 

LCM = 8\times3\times5=120

The prime factors of both 495 and 220 can be displayed using prime factor trees. 

 

So, the factorisation of 220 is,

220=2\times2\times5\times11

 

and the factorisation of 495 is,

 

495=3\times3\times5\times11

 

Now, we will draw a Venn diagram with one circle containing the factors of 495 and the other containing the factors of 220. Any prime factors shared by these two numbers are to be placed in the intersection.

 

495=3\times3\times\cancel5\times\cancel11

220=2\times2\times\cancel5\times\cancel11

 

The HCF can be calculated by multiplying the numbers in the intersection together,

 

\text{HCF }=5\times11=55

 

Finally, we find the LCM by multiplying all the numbers in the Venn diagram together,

 

\text{LCM }=3\times3\times5\times11\times2\times2=1,980

First, we have to find the prime factorisation of 32, 152 and of 600:

 

Prime factors of 32=2\times2\times2\times2\times2

Prime factors of 152= 2\times2\times2\times19

Prime factors of 600= 2\times2\times2\times3\times5\times5

 

Then we can place each prime factor in the correct circle in the Venn diagram. Any common factors should be placed in the intersections of the circles.

 

The highest common factor (HCF) is found by multiplying together the numbers in the intersection of all three of the circles.

HCF2\times2\times2=8

 

The lowest common multiple (LCM) is found by multiplying together the numbers from all sections of the circles.

 

LCM2\times2\times2\times2\times2\times3\times5\times5\times19=45600

Related Topics

MME

Types of Numbers Worksheets, Questions and Revision

Level 1-3GCSEKS3

Worksheet and Example Questions

Site Logo

(NEW) Prime Factors, HCF, LCM - Exam Style Questions - MME

Level 4-5 GCSE KS3NewOfficial MME

Drill Questions

Site Logo

HCF LCM Product Of Primes - Drill Questions

Level 4-5 GCSE KS3
Site Logo

BIDMAS and Prime Factors - Drill Questions

Level 4-5 GCSE KS3
Site Logo

LCM HCF Factors and Prime Numbers - Drill Questions

Level 4-5 GCSE KS3
Site Logo

Factors Multiples Primes - Drill Questions

Level 4-5 GCSE KS3

You May Also Like...

GCSE Maths Revision Cards

Revise for your GCSE maths exam using the most comprehensive maths revision cards available. These GCSE Maths revision cards are relevant for all major exam boards including AQA, OCR, Edexcel and WJEC.

£8.99
View Product

GCSE Maths Revision Guide

The MME GCSE maths revision guide covers the entire GCSE maths course with easy to understand examples, explanations and plenty of exam style questions. We also provide a separate answer book to make checking your answers easier!

From: £14.99
View Product