# Multiples, factors and prime factors

## In a nutshell

Multiples and factors are both very useful applications of multiplication and division. Furthermore, numbers can be broken down into their prime factors, which has useful applications.

## Multiples and factors

### Definitions

A **multiple** of a number is an integer multiplied by the number.

A **factor** of a number divides into the number without any remainders.

##### Examples

*$20$ is a multiple of $5$ because $20=4\times5.$**$4$ is a factor of $100$ because $100\div4=25$.**$18$ is **not** a multiple of $10$ because $18$ is not in the $10$ times tables.*- $8$
*is **not** a factor of $12$ because $12\div8=1$ remainder $4$.*

### Finding factors of a number

To find all the factors of a number, follow this procedure.

#### Procedure

- Write the number as $1\times$ itself.
- Move onto the next number ($2$), and find the pair that multiplies by $2$ by writing the number as $2\times \_$ (if applicable).
- Move onto the next number, and repeat the process until you get repeated pairs.

- The factors are all the numbers that have a pair where both numbers are integers.

##### Example 1

*Find all the factors of $18$.*

*Start from $1$:*

$18=1\times18$

*Hence, $1$ and $18$* *are factors.*

*Continue in the same fashion with $2,3,4...$* *until you get a repeated pair.*

*$18=2\times9$*

*$18=3\times6$*

*$18=4\times4.5$*

*$18=5\times3.6$*

*$18=6\times3$*

*This is a repeated pair ($3\times6$ and $6\times3$). So, stop here.*

*The factors of ** *$18$* are *$\underline{1,2,3,6,9,18}$*.*

## Prime factors

A **prime factor** of a number is a factor that is also *prime. *

### Prime factor decomposition

Every positive integer can be broken down into a *unique *product of prime factors. This means to write a number as a prime number or numbers being multiplied together. For example, the number $18$ can be written as $3\times3\times2$, or $3^2\times2$. This is called the *prime factorisation** * of a number. To find a number's *prime factorisation*, you can use a **factor tree**.

#### procedure

- Write down the number at the top. Find two numbers that multiply to give the number.

- Write down these two factors, branching off to the bottom left and bottom right of the original number.

- Proceed similarly with each individual factor, until each "branch" results in a prime factor.

- Circle each prime factor, and write them all multiplied by each other. This is the prime factorisation.

##### Example 2

*Find the prime factorisation of $180$.*

*Write $180$ on the top, and find two numbers that multiply to give $180$:*

$180=18\times10$

*Write $18$ and $10$* *beneath $180$*, *and do the same for each factor.*

*For $10$:*

$10=\raisebox{.5pt}{\textcircled{\raisebox{-.6pt} {5}}}\times\raisebox{.5pt}{\textcircled{\raisebox{-.6pt} {2}}}$

*Note that $5$ and $2$ are both prime numbers, so circle them and move on to $18$.*

*For $18$:*

$18=9\times\raisebox{.5pt}{\textcircled{\raisebox{-.6pt} {2}}}$

*Note that $2$ is prime, so circle it and move on to $9$.*

*For $9$:*

$9=\raisebox{.5pt}{\textcircled{\raisebox{-.6pt} {3}}}\times\raisebox{.5pt}{\textcircled{\raisebox{-.6pt} {3}}}$

$3$ *is prime, so circle both the threes.*

*Now, write down all the circled numbers (these are the prime factors) multiplied by each other:*

*$180=2\times2\times3\times3\times5$*

*Write the repeated factors as indices:*

*$\underline{180=2^2\times3^2\times5}$*

**