# Integer Divisor Results

## Theorem

Let $m, n \in \Z$ be integers.

Let $m \divides n$ denote that $m$ is a divisor of $n$.

The following results all hold:

### One Divides all Integers

 $\displaystyle 1$ $\divides$ $\displaystyle n$ $\displaystyle -1$ $\divides$ $\displaystyle n$

### Integer Divides Itself

$n \divides n$

### Integer Divides its Negative

 $\displaystyle n$ $\divides$ $\displaystyle -n$ $\displaystyle -n$ $\divides$ $\displaystyle n$

### Integer Divides its Absolute Value

 $\displaystyle n$ $\backslash$ $\displaystyle \left \lvert {n}\right \rvert$ $\displaystyle \left \lvert {n}\right \rvert$ $\backslash$ $\displaystyle n$

where:

$\left\lvert{n}\right\rvert$ is the absolute value of $n$
$\backslash$ denotes divisibility.

### Integer Divides Zero

$n \divides 0$

### Divisors of Negative Values

$m \mathrel \backslash n \iff -m \divides n \iff m \divides -n \iff -m \divides -n$