GCSE Tutoring Programme
"Our chosen students improved 1.19 of a grade on average - 0.45 more than those who didn't have the tutoring."
In order to access this I need to be confident with:
Prime factors Factor trees Factors and multiplesThis topic is relevant for:
Here we will learn about the highest common factor including how to calculate the highest common factor, use the highest common factor, and recognise when to calculate the highest common factor to answer complex worded problems.
There are also highest common factor worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if youβre still stuck.
The highest common factor (HCF) or greatest common factor is the largest integer that two or more numbers can be divided by.
Highest meaning largest number or greatest.
Common meaning shared between two or more numbers.
Factor meaning an integer that a whole number can be divided by (a divisor).
There are several interchangeable terms you should be aware of:
HCF represents the highest common factor, GCF represents the greatest common factor, and GCD represents the greatest common divisor. These are all the same!
E.g.
Find the HCF of
Let’s start by writing the factors of 4 and 6,
We can see that the highest number that occurs in each list is
hence for the numbers
NOTE:
Calculating the highest common factor becomes more complicated for larger numbers as listing all the factors of each number is very time consuming.
To make it easier we can utilise the prime factors of the numbers.
The fundamental theorem of arithmetic states that each number greater than
This means that every number has a unique set of prime factors.
We can use prime factors and a Venn diagram to calculate the highest common factor.
To do this we need to write out the prime factorisation of each number fully (without any powers) and then put the numbers into the Venn diagram by looking for pairs.
Here, we have the prime factors of
Here, the prime factors of 2 and 3 occur for both 12 and 30 so their product is the highest common factor.
The highest common factor for
The reason why the intersection contains the highest common factor is because there are no other common prime factors in this set of numbers, otherwise they would be in the intersection.
You must remember to write every factor out separately into the Venn diagram.
If you have more than
In order to calculate the highest common factor for two or more numbers:
Get your free highest common factor worksheet of 20+ factors questions and answers. Includes reasoning and applied questions.
DOWNLOAD FREEGet your free highest common factor worksheet of 20+ factors questions and answers. Includes reasoning and applied questions.
DOWNLOAD FREEHighest common factor is part of our series of lessons to support revision on factors, multiples and primes. You may find it helpful to start with the main factors, multiples and primes lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:
Calculate the highest common factor of
2Write all the prime factors into the Venn diagram for each number
3Multiply the prime factors in the intersection to find the HCF
HCF =
Calculate the highest common factor of
State the product of prime factors for each number, not in index form
Write all the prime factors into the Venn diagram for each number
Multiply the prime factors in the intersection to find the HCF
HCF =
Given that
State the product of prime factors for each number, not in index form
To state the product of prime factors of
Write all the prime factors into the Venn diagram for each number
Multiply the prime factors in the intersection to find the HCF
HCF =
Calculate the highest common factor of
State the product of prime factors for each number, not in index form
Write all the prime factors into the Venn diagram for each number
Multiply the prime factors in the intersection to find the HCF
HCF =
Calculate the highest common factor of
State the product of prime factors for each number, not in index form
Write all the prime factors into the Venn diagram for each number
Multiply the prime factors in the intersection to find the HCF
Here there are no prime factors in the intersection, so the highest common factor is equal to
Sue sells apples in baskets. On Thursday, she will receive a delivery of
State the product of prime factors for each number, not in index form
Write all the prime factors into the Venn diagram for each number
Multiply the prime factors in the intersection to find the HCF
HCF
So there will be
A very common misconception is mixing up the highest common factor with the lowest common multiple.
Remember:
Factors are composite numbers that are split into smaller factors
Multiples are composite numbers that are multiplied to make larger multiples
It is possible to write prime factors into a Venn diagram with their associated exponent or power. This becomes an issue when the powers are not correctly interpreted.
Take for example the numbers
This means that we have common factors of
The correct answer is
The remaining factor of
The Venn diagram would therefore look like this:
It is recommended that you write out each product of prime factors without using index form as this will make writing the numbers into the Venn diagram easier.
1. Calculate the highest common factor of 22 and 60 .
2. Calculate the highest common factor of 21 and 63 .
3. Calculate the highest common factor of 90 and 135 .
4. Calculate the highest common factor of 12 , 30 and 48 .
5. Calculate the highest common factor of 21 and 58 .
6. Josh is grouping 3 different coloured writing pens. He has 18 blue, 16 black, and 28 red. Each group must contain the same amount of each colour pen. How many groups can Josh make, so that every pen has been grouped?
We need to calculate the HCF of 18, 16 and 28. \\18=2 \times 3 \times 3\\ 16=2 \times 2 \times 2 \times 2\\ 28=2 \times 2 \times 7 \\ HCF=2
Therefore Josh can make 2 groups of pens with the same number of each colour in each group.
1. The highest common factor of a and b is 5 . The lowest common multiple of a and b is 30 . State the values of a and b .
(4 marks)
(1)
30 \div 5=6
(1)
6= 2 \times 3
(1)
2\times 5=10 \text{ and } 5 \times 3=15a=10 and b = 15
(1)
(accept the alternative solution: a=15 and b=10 )
2.
(a) Calculate the highest common factor of the two numbers a=16 g^{2} h^{2} \text { and } b=28 g^{3} h
(b) Let g=3 and h=5
(i) Calculate the values of a and b
(ii) Hence or otherwise, calculate the value of the highest common factor when g=3 and h=5
(8 marks)
(1)
HCF \text { of } 16 \text { and } 28=2 \times 2=4(1)
HCF \text { of } g^{2} h^{2} \text { and } g^{3} h=g^{2} h(1)
HCF \text { of } 16 g^{2} h^{2} \text { and } 28g^{3}h =4g^{2}h(1)
\text { b)(i) } a = 16 \times 3 \times 3 \times 5 \times 5 = 3600(1)
b = 28 \times 3 \times 3 \times 3 \times 5 = 3780(1)
\text { b)(ii) } 4 \times 3 \times 3 \times 5(1)
180(1)
3. A school is conducting a research project. Year 11 classes will be split into small groups with the same proportion of students from each class.
What is the maximum number of groups possible?
(3 marks)
(1)
28=2 \times 2 \times 7 . \text { and } 36=2 \times 2 \times 3 \times 3Β(1)
\mathrm {HCF} (32,28 \text { and } 36)=4(1)
You have now learned how to:
Prepare your KS4 students for maths GCSEs success with Third Space Learning. Weekly online one to one GCSE maths revision lessons delivered by expert maths tutors.
Find out more about our GCSE maths tuition programme.