# Prime Numbers

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## Prime Numbers

A prime number is a special type of number that can only be divided by $1$ and itself; they only appear in their own times table.

## Skill 1: Factors and Multiples

A multiple of a number is just a number that appears in its times table.

A factor is a number by which another number can be divided. A pair of factors can be multiplied together to give the original number.

For example, if we want to list the factors of the number $12$, we need to find the numbers that $12$ can be divided by. We can write them out as pairs:

$1 \times 12 = 12$

$2 \times 6 = 12$

$3 \times 4 = 12$

We can say that $1$, $2$, $3$, $4$, $6$ and $12$ are all factors of $12$. Another way of saying this is that $12$ is a multiple of each of its factors.

Level 1-3 GCSE KS3

## Skill 2: Understanding Prime Numbers

Another way of describing prime numbers is that they are numbers with only two positive factors. There are some important facts you should know about prime numbers:

1. The number $\textcolor{black}1$ is not a prime number – it has only one positive factor (itself!)
2. The number $\textcolor{red}2$ is the only even prime number
3. Except for $\textcolor{red}2$ and $\textcolor{red}5$, all prime numbers end in $\textcolor{red}1$, $\textcolor{red}3$, $\textcolor{red}7$ or $\textcolor{red}9$
4. Not all numbers ending in $\textcolor{black}1$, $\textcolor{black}3$, $\textcolor{black}7$ and $\textcolor{black}9$ are prime
5. There is no pattern in the series of prime numbers

The prime numbers between $\textcolor{black}1$ and $\textcolor{black}{50}$ are displayed below.

Level 1-3 GCSE KS3

## Skill 3: Finding Prime Numbers

You may be asked to show whether or not a number is prime. There are a couple of things to bear in mind that make this process a lot easier:

• Ignore any numbers that end in $\textcolor{blue}0$, $\textcolor{blue}2$, $\textcolor{blue}4$, $\textcolor{blue}6$ and $\textcolor{blue}8$ – these are even numbers and are all divisible by $\textcolor{blue}2$ (except for $0$ itself!)
• If you think a number is prime, and it does not divide exactly by $\textcolor{red}3$ or $\textcolor{red}7$ then it is a prime number. This applies to all prime numbers less than $120$
• For prime numbers larger than $120$, you may have to try dividing them by many other numbers
Level 1-3 GCSE KS3

## Skill 4: Prime Factor Trees

Prime numbers are the ‘building blocks’ of other larger numbers. We can divide a number into it’s factors and keep dividing these factors by their factors until we are left with nothing but prime numbers – this is called prime factorisation. We use factor trees to find the prime factorisation of a number.

Example: find the prime factorisation of $126$

First, recognise that $126$ is an even number so it is divisible by $2$:

$126\div2 = 63$

We are left with $2$ and $63$. $\textcolor{red}2$ is a prime number, so we circle it.

Next, we can divide $63$ by $3$:

$63 \div 3 = 21$

We are left with $3$ and $21$. $\textcolor{red}3$ is a prime number, so we circle it.

Next, we can divide $21$ by $3$:

$21\div3=7$

We are left with $3$ and $7$ which are both prime numbers, so we circle them both.

There are no more ways we can divide the factors as we are left with prime numbers only. We can now write the prime factorisation as:

$2\times3\times3\times7=126$

We can convert this to index form by grouping the numbers that are the same into powers:

$2\times3^2 \times7 = 126$

Level 1-3GCSEKS3

## Example 1: Finding a Prime Number

Determine whether the following numbers are prime:

$25, 31, 57, 84$

The number $25$ is divisible by $5$ so it is not prime: $25 \div 5 = 5$

The number $31$ cannot be divided by any number other than $1$ and itself, so it is prime.

The number $57$ is divisible by $3$ so it is not prime: $57 \div 3 = 19$

The number $84$ is divisible by $2$ so it is not prime: $84 \div 2 = 42$

Level 1-3 GCSE KS3

## Example Questions

To show that a number is not prime, we just need to find a pair of factors other than $1 \times$ itself.

Another pair of factors of $39$ are:

$13\times3=39$

Therefore $39$ is not a prime number.

First, cross out all the even numbers:

$\cancel{20},21,\cancel{22},23,\cancel{24},25,\cancel{26},27,\cancel{28},29$

We are now left with the following numbers:

$21,23,25,27,29$

We can cross out $25$ as all prime numbers end in either $1$, $3$, $7$ or $9$.

We are now left with:

$21,23,27,29$

$21\div3 = 7$ so $21$ is not a prime number.

$27\div 3 = 9$ so $23$ is not a prime number.

$23$ and $29$ have only two factors each;

$1\times23 = 23$ and $1 \times 29 = 29$

Therefore $23$ and $29$ are the only prime numbers in the list.

We need to list the first $5$ numbers in the $8$ times table. These are:

$8,16,24,32,40$

It is easiest to list factors in pairs:

$1\times48 = 48$

$2\times24 = 48$

$3\times16=48$

$4\times12=48$

$6\times8=48$

Therefore the factors of $48$ are:

$1,2,3,4,6,8,12,16,24,48$

There are many factor pairs of $72$, but the easiest way to find the prime factorisation is to start by dividing by $2$:

We can carry on dividing the factors by $2$ until we are left with an odd number ($9$), at which point we can divide by $3$.

We are left with the prime factorisation:

$2\times2\times2\times3\times3 = 2^3 \times 3^2 = 72$

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