# Integer Divisor Results/Integer Divides its Negative

## Theorem

Let $n \in \Z$ be an integer.

Then:

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

where $\divides$ denotes divisibility.

## Proof

From Integers form Integral Domain, the integers are an integral domain.

Hence we can apply Product of Ring Negatives:

$\forall n \in \Z: \exists -1 \in \Z: n = \paren {-1} \times \paren {-n}$
$\forall n \in \Z: \exists -1 \in \Z: -n = \paren {-1} \times \paren n$

$\blacksquare$