Albert Teen YOU ARE LEARNING:  Highest Common Factors  # 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.

1

What is a factor of a number? 2

Which of these are factors of $36$? Select all that apply. 3

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 $51$? 4

Finding the highest common factor, or HCF, of two numbers can help when working with fractions.

We find the HCF using prime factorisation.

1

Let's try an example.

What is the HCF of $30$ and $75$?

2

First we find the prime factorisation of each number.

This is the prime factorisation tree for $30$. 3

Using a prime factorisation tree, find the prime factors of $75$.  1

We now have the prime factorisations of both $30$ and $75$. One common factor is $3$, what is the other?  2

We have identified that $3$ and $5$ are common prime factors of $30$ and $75$. How can we find the highest common factor?  3

The common prime factors are highlighted, what is the highest common factor of $30$ and $75$?  4

Well done! You have found the answer!

The HCF of $30$ and $75$ is $15$. 1

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 $39$.

2

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.  3

Now we have all the factors of the two numbers, what is the HCF of $27$ and $39$?  4

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! 1

Quick recap - which option describes the HCF? 2

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 1

Let's try a harder example: finding the HCF of three numbers.

Let's find the HCF of $200$, $60$ and $40$.

2

The prime factorisation of $200$ is $2^3\times 5^2$. What is the prime factorisation of $60$?  3

We need to find our final prime factorisation of $40$, what is this one?  4

We now have the prime factorisations for all three numbers.

We need to compare them and find which are common to all three numbers. 5

Which two prime numbers appear in all three? Separate your answers with a comma.  6

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?  7

Now let's find out HCF of $200$, $60$ and $40$. The common factors are $2^2$ and $5$, what is the HCF?  8

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. Summary!

1

A factor is a number which divides into another number exactly, leaving no remainder, decimal or fraction.

$5$ is a factor of $20$ because $20\div 5=4$

$8$ is not a factor of $20$ because $20\div 8=2.5$

2

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 $75\div 3=25$ and $27\div 3=9$

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Numbers can have more than one common factor.

Finding the highest common factor of two numbers can help when working with fractions.

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Use prime factorisation of the numbers to find the highest common factor.

The prime factors in common multiply together to give the HCF.

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To find the HCF of $84$ and $132$, find the prime factorisation of both numbers.

$84=2^2\times 3\times 7$ and $132=2^2\times 3\times 11$ The HCF is the product of the common factors which is $2^2\times 3=12$

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Where a common prime factor has a power

choose the lowest power to find the HCF.

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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.

8

$12$ 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 $6$.