Back to AQA Multiples & Factors (F) Home
1.4 F) Highest Common Factor & Lowest Common Multiple – Part 1
1.4 F) Highest Common Factor & Lowest Common Multiple – Part 1
Before you work through this section, make sure that you have looked at the section about writing a number as a product of its prime factors. This is because this section builds on what was covered in that section. Click here to be taken through to the section about prime factors.
Highest Common Factor
The highest common factor is the highest number that goes into two (or more) numbers. There are two ways that we are able to obtain the highest common factors for two or more numbers.
Personally, I think that the second method is the better method because there is more chance of missing out a factor when using the first method. Also, the second method is better at finding the lowest common multiple for two numbers and many questions in the exam will ask you to find both the highest common factor and the lowest common multiple.
I will go through both methods and you can choose the method that you prefer.
The highest common factor is the highest number that goes into two (or more) numbers. There are two ways that we are able to obtain the highest common factors for two or more numbers.
- One method is to write down all of the factors for both of the numbers and find the highest factor that goes into both of the numbers.
- The second method is to use factor trees
Personally, I think that the second method is the better method because there is more chance of missing out a factor when using the first method. Also, the second method is better at finding the lowest common multiple for two numbers and many questions in the exam will ask you to find both the highest common factor and the lowest common multiple.
I will go through both methods and you can choose the method that you prefer.
Example 1
What is the highest common factor for 12 and 28?
Method 1 – Writing Out All Factors
I am going to answer this question by using the first method, which was to write down all of the factors for both numbers and then find the highest factor that appears in both of the lists. It is best to use factor pairs to find all of the factors of a number. This is because by using factor pairs, there is less chance of missing potential factors out. I am going to do this for 12 and 28.
Factor pair for 12:
What is the highest common factor for 12 and 28?
Method 1 – Writing Out All Factors
I am going to answer this question by using the first method, which was to write down all of the factors for both numbers and then find the highest factor that appears in both of the lists. It is best to use factor pairs to find all of the factors of a number. This is because by using factor pairs, there is less chance of missing potential factors out. I am going to do this for 12 and 28.
Factor pair for 12:
All factors of 12: 1, 2, 3, 4, 6 and 12
Factors of 28:
Factors of 28:
All factors of 28: 1, 2, 4, 7, 14 and 28
We now compare the two lists of factors.
We now compare the two lists of factors.
The highest number that appears in both of the lists is 4, which means that 4 is the highest common factor for 12 and 28.
Method 2 – Factor Trees
In this method for are going to express both of the numbers as a product of their prime factors. After both of the numbers are expressed as a product of their prime factors, we can then find the highest common factor by multiplying the factors that appear in both of the product of their prime factor calculations.
The prime factor trees for 12 and 28 are given below.
Method 2 – Factor Trees
In this method for are going to express both of the numbers as a product of their prime factors. After both of the numbers are expressed as a product of their prime factors, we can then find the highest common factor by multiplying the factors that appear in both of the product of their prime factor calculations.
The prime factor trees for 12 and 28 are given below.
It is easier to see common factors between the two numbers by expressing the prime factors with no indices/ powers.
The common prime factors between 12 and 28 are two 2’s. We multiply the common factors together to obtain the highest common factor for 12 and 28. Therefore, the highest common factor is 4.
This is the same answer as we obtained using the other method.
Example 2
What is the highest common factor for 42 and 63?
I am going to answer this question using both methods.
Method 1 – Writing Out All Factors
In this method, we write down all of the factors of the two numbers and then find the highest factor that is a factor in both of the lists. I am going to write the factors down as pairs so that I do not miss any factors.
Let’s start by finding the factor pairs for 42:
What is the highest common factor for 42 and 63?
I am going to answer this question using both methods.
Method 1 – Writing Out All Factors
In this method, we write down all of the factors of the two numbers and then find the highest factor that is a factor in both of the lists. I am going to write the factors down as pairs so that I do not miss any factors.
Let’s start by finding the factor pairs for 42:
All factors of 42: 1, 2, 3, 6, 7, 14, 21 and 42
The factor pairs of 63 are:
The factor pairs of 63 are:
All the factors of 63: 1, 3, 7, 9, 21 and 63
We now compare the two lists of factors.
We now compare the two lists of factors.
The highest factors that appears in both lists is 21. Therefore, the highest common factor for 42 and 63 is 21.
Method 2 – Factor Trees
In this method we are going to express both of the numbers as a product of their prime factors. We are then able to find the highest common factor by multiplying the common factors.
The factor trees for these two numbers are shown below.
Method 2 – Factor Trees
In this method we are going to express both of the numbers as a product of their prime factors. We are then able to find the highest common factor by multiplying the common factors.
The factor trees for these two numbers are shown below.
The two numbers as a product of their prime factors are shown below (it is easier to not use any powers).
There are two common prime factors that appear in both; 3 and 7. We find the highest common factor by multiplying the common prime factors together. Therefore, we multiply 3 and 7, which results in the highest common factor being 21.
This is the same answer as we obtained when we used the other method.
End Note
It is a good idea to try both of the methods for some of the questions in this quiz. After trying both methods, you can then decide which method you prefer.
It is a good idea to try both of the methods for some of the questions in this quiz. After trying both methods, you can then decide which method you prefer.