Albert Teen YOU ARE LEARNING:  Prime Factorisation  # Prime Factorisation ### Prime Factorisation

AKA the Unique Factorisation Theorem, we can split any number down into a product of prime numbers, where each number has one unique combination!

1

A factor is a number that divides into another number exactly. Which of these numbers is a factor of $28$? 2

Because $28\div 7=4$ we know that $7$ is a factor of $28$.

It also shows that $4$ is a factor of $28$ because $28\div 4=7$.

1

We have seen that $4$ and $7$ are factors of $28$. A prime factor, is a factor that is a prime number. Which one of these is a prime factor of $28$? 2

We have found two factors of $28$ and found that $7$ is a prime factor. What other factors are there are of $28$? Select all that apply. 3

We now have all the factors of $28$.

They are $1$, $2$, $4$, $7$, $14$ and $28$.

1

Finding all the factors of a number can be quite tedious.

We can use a process called prime factorisation to help.

2

We have seen that all the factors of $28$ are $1$, $2$, $4$, $7$, $14$ and $28$. Which of these are prime? Select two and separate your answer with a comma.  3

We have now found the prime factors of $28$, they are $2$ and $7$. Which calculation uses those factors to make $28$? 4

We have found the prime factorisation of $28$ which is $2\times 2\times 7$.

Note that sometimes a prime factor is repeated.

5

Prime number factorisation helps us in other areas of maths.

If can help us to find common multiples and factors which in turn help working with fractions.

1

Let's use a prime factor tree to find the prime factors of a number.

We will identify the prime numbers as they appear.

2

Starting with $28$, we find a factor. We start with $4$ and put that in the next tier. What number will replace the $?$?  3

We now look at our factors and determine if either of them is prime.

We know that $7$ is prime, so we circle it to show it is the end of that branch. 4

We now focus on the number $4$. What number is a factor of $4$ that can replace both the question marks?  5

We have replaced the $?$ with the number $2$.

We know that $2$ is prime so circle them both. 6

We now have the prime factorisation of the number $28$.

Take the circled prime numbers and multiply them together: $2\times 2\times 7=28$ 1

Let's find the prime factorisation of $16$. If the first factor is $2$, what number completes this stage of the tree?  2

We've added $8$ which also has a factor of $2$. What completes this stage of the tree?  3

Circle the prime numbers as you go. What number completes this tree?  4

This is our completed tree for the prime factorisation of $16$.

We bring this together by multiplying the prime factors so we have $2\times 2\times 2\times 2=16$. 5

It's not the only way to draw a factor tree for $16$

Instead, we could start by splitting $16$ into $4 \times 4$. 6

What's the prime factorisation now?  7

The two trees are different but the final answer is still the same.

Every composite number has a unique prime factorisation. 1

We have seen that the prime factorisation of $16$ is $2\times 2\times 2\times 2$.

We can shorten this using the fact that powers of a number are repeated multiplication.

2

There are four $2s$ in the expression $2\times 2\times 2\times 2$. Keeping $2$ as the base number, what is the power we need to use? 3

This gives an alternative way of writing the prime factorisation of $16$.

$2^4=16$

Quick recap: what is prime factorisation and why is it helpful? Select all the options that apply. What is the prime factorisation of $36$? Take some time to work this out if you need to We have seen that the prime factorisation of $36$ is $2\times 2\times 3\times 3$. What is an alternative way of writing this? Final question! Use prime factorisation to express $54$ as the product of its prime factors. Keep the factors in number order in your answer. Summary!

1

A factor is a number which divides exactly into another number with no remainder or decimal answer.

$4$ is a factor of $12$ as $12\div 4=3$

$5$ is not a factor of $12$ as $12\div 5=2.4$

2

A prime factor is a factor that is a prime number.

$4$ is a factor of $12$ but it is not a prime factor

$3$ is a prime factor of $12$ 3

Every number has a unique prime factorisation

The prime factorisation of $12$ is $2\times 2\times 3$ or $2^2\times 3$. 4

Use a prime factor tree to find the prime factorisation of a number.

Prime factorisation of $28$ is $2\times 2\times 7$ or $2^{2}\times7$ 