# What's a multiple of 4? What does that mean?

Apr 17, 2016

A multiple of $4$ is any number which can be written in the form $4 n$ where $n$ is an integer.

#### Explanation:

In general, given two values $a$ and $b$, we say that $b$ is a multiple of $a$ if $b = n a$ for some integer $n$.

For example, $- 4 , 0 , 4 , 12 ,$ and $100$ are all multiples of $4$ because

$- 4 = 4 \times - 1$
$0 = 4 \times 0$
$4 = 4 \times 1$
$12 = 4 \times 3$
$100 = 4 \times 25$

On the other hand, values such as $1$ or $10$ are not multiples of $4$, as

$1 = 4 \times \frac{1}{4}$
$10 = 4 \times \frac{5}{2}$

$\frac{1}{4}$ and $\frac{5}{2}$ are not integers.

It is generally correct to state that $b$ is a multiple of $a$ if $\frac{b}{a}$ is an integer, although this requires $a \ne 0$.

Intuitively, it makes sense to call these values multiples, as they are the values which can be obtained by adding or subtracting your original value multiple times. A multiple of $4$ is a number which can be written as
$4 + 4 + 4 + \ldots + 4$
or
$4 - 4 - 4 - \ldots - 4$.
Note that this is equivalent to the above definitions.