Integer Divisor Results/Integer Divides its Negative

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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:

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

and Product with Ring Negative:

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

$\blacksquare$


Sources