# Multiples, factors and prime factors

## In a nutshell

Every number is either a prime number, or a multiple of other factors.

## Multiples, factors and primes

**Multiple -** A multiple of a number is any number that can be formed as a product of that number and another number.

**Factor -** A factor of a number is any number that divides the number with no remainder.**Prime number -** A prime number is a number which has only two distinct factors, $1$ and itself.

##### Example 1

*What are the factors of $36$?*

*$36 = 1 \times 36$*

*$36 = 2 \times 18$*

*$36 = 3 \times 12$*

*$36 = 4 \times 9$*

*$36 = 6\times 6$*

*Hence the factors of $36$ are $\underline{1,2,3,4,6,9,12,18,36}$.*

##### Example 2

*Is $162$ a multiple of $27$?*

*$162 \div 27 = 6$*

*Yes,** *$162$* is a multiple** of *$27$*, as the quotient (*$6$*) is a whole number and there is no remainder.*

## Factor trees

You can find the prime factorisation of a number using a factor tree.

#### Procedure

- Write the number at the top and find two numbers which multiply to make the number.

- Write the pair of factors underneath the original number, "branching" them off to the left and right of the original number.

- If one, or both, of these factors are not prime, repeat this process until the end of every "branch" is a prime number.

- Write the numbers at the bottom of each branch, multiplied by one another. This is the prime factorisation.

##### Example 3

*Use a factor tree to find the prime factorisation of $273$.*

$\quad\quad\quad\space{\begin{aligned} &\space \space \,\,273\\\ &\,\,\swarrow\searrow& \\&\textcircled{3}\,\,\,\,\, \quad91 \\&\quad\,\,\swarrow\searrow\\&\space\space\space\,\,\textcircled7 \,\,\,\,\quad\textcircled{13} \\\end{aligned}}$

*Hence the prime factorisation of $273$ is $\underline{273 = 3 \times 7 \times 13}$.*

**Note: **If the original number is, itself, a prime number then you already have the number in its prime factorisation form.