# Prime numbers and prime factorisation

## In a nutshell

Multiplying numbers together forms a product, and this resulting number has factors. Some numbers only have two factors, and these are prime numbers.

## Prime numbers

### Definition

A prime number is a number that is greater than $1$ and has exactly $2$ factors: $1$ and itself.

The sequence of prime numbers goes as follows:

$2,3,5,7,11,13,17,19,23,29,31,37,41,43,47, ...$

## Prime factorisation

The process of prime factorisation allows you to write any number as a product of its prime factors.

#### procedure

- Find the smallest prime number which divides the number.

- Write the number as a product of this prime number and the quotient.

- Repeat this process with the quotient until the quotient is a prime number.

- The prime factorisation is the product of all the prime numbers.

##### Example 1

*What is the prime factorisation of $54$?*

*$54 = 27 \times 2$*

*$27 = 9 \times 3$*

*$9 = 3\times 3$*

*The factors can also been shown using a factor tree:*

*$\begin {array}{cccccccccc}&& 54 &&&\\& \swarrow && \searrow \\\\ \textcircled{2} &&&& 27 \\&&& \swarrow && \searrow \\\\ &&\textcircled{3} &&&& 9 \\&&&&& \swarrow && \searrow \\\\ &&&&\textcircled{3} &&&& \textcircled{3} \\\end {array}$*

**

*Hence, the prime factorisation of $54$ is *$\underline{ 54 = 2 \times 3 \times 3 \times 3= 2 \times 3^{3}}$.