Negative Number is Congruent to Modulus minus Number

Theorem

 * $\forall m, n \in \Z: -m \equiv n - m \pmod n$

where $\bmod n$ denotes congruence modulo $n$.

Proof
Let $-m = r + k n$.

Then $-m + n = r + \paren {k + 1} n$

and the result follows directly by definition.

Also see

 * Wilson's Theorem, where this is used:
 * $-1 \equiv \paren {p - 1}! \pmod p \iff \text {$p$ is prime}$