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Highest Common Factors
Highest Common Factors
The highest common factor is the highest number which is a factor of two other numbers. We can use prime factorisation to find it.
What is a factor of a number?
Which of these are factors of 36? Select all that apply.
You can select multiple answers
A common factor of two numbers is a factor that divides into both numbers without leaving a remainder. Which of these is a common factor of 36 and 51?
Finding the highest common factor, or HCF, of two numbers can help when working with fractions.
We find the HCF using prime factorisation.
Let's try an example.
What is the HCF of 30 and 75?
First we find the prime factorisation of each number.
This is the prime factorisation tree for 30.
Using a prime factorisation tree, find the prime factors of 75.
We now have the prime factorisations of both 30 and 75. One common factor is 3, what is the other?
We have identified that 3 and 5 are common prime factors of 30 and 75. How can we find the highest common factor?
The common prime factors are highlighted, what is the highest common factor of 30 and 75?
Well done! You have found the answer!
The HCF of 30 and 75 is 15.
When the numbers are smaller, we can list the factors to find the highest common factor.
Let's find the highest common factor of 27 and 39.
We can list the factors of 39, they are 1, 3, 13 and 39. What are the factors of 27? List them in order and separate your answers with a comma.
Now we have all the factors of the two numbers, what is the HCF of 27 and 39?
The HCF of 27 and 39 is 3.
With this method, when listing the factors we just find the highest one, there is no multiplying needed!
Quick recap - which option describes the HCF?
How can you find the HCF by listing the factors of each number?
What is the highest common factor of 15 and 35?
Fred says that 8 is the HCF of 24 and 48. Is he correct? If not, what is the HCF of 24 and 48?
We've seen that the HCF of 24 and 48 is 24.
This shows that sometimes the HCF of two numbers is the smaller of the two numbers.
What is the HCF of 36 and 180?
You may need some time to work this out
Let's try a harder example: finding the HCF of three numbers.
Let's find the HCF of 200, 60 and 40.
The prime factorisation of 200 is 23×52. What is the prime factorisation of 60?
We need to find our final prime factorisation of 40, what is this one?
We now have the prime factorisations for all three numbers.
We need to compare them and find which are common to all three numbers.
Which two prime numbers appear in all three? Separate your answers with a comma.
Be careful with this one! We identified that the common prime factors are 2 and 5, but look more closely at the 2. What is the actual common factor here?
Now let's find out HCF of 200, 60 and 40. The common factors are 22 and 5, what is the HCF?
Well done! The HCF of 200, 60 and 40 is 20.
Where a common prime factor has a power, we choose the lowest power to find the HCF.
A factor is a number which divides into another number exactly, leaving no remainder, decimal or fraction.
5 is a factor of 20 because 20÷5=4
8 is not a factor of 20 because 20÷8=2.5
A common factor of two numbers is the same factor that divides exactly into both numbers.
3 is a common factor of 75 and 27 because 75÷3=25 and 27÷3=9
Numbers can have more than one common factor.
Finding the highest common factor of two numbers can help when working with fractions.
Use prime factorisation of the numbers to find the highest common factor.
The prime factors in common multiply together to give the HCF.
To find the HCF of 84 and 132, find the prime factorisation of both numbers.
84=22×3×7 and 132=22×3×11 The HCF is the product of the common factors which is 22×3=12
Where a common prime factor has a power
choose the lowest power to find the HCF.
For smaller numbers it may be easier simply to list all the factors of both.
The HCF is the highest factor that appears in both lists.
12 has factors 1, 2, 3, 4, 6 and 12
18 has factors 1, 2, 1, 1, 1 and 1
The highest factor in both lists is 6 so the HCF of 12 and 18 is 6.