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 $\underline{ 54 = 2 \times 3 \times 3 \times 3= 2 \times 3^{3}}$ .