Negative Number is Congruent to Modulus minus Number

Theorem

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

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

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

Then $$-m + n = r + \left({k + 1}\right) n$$

and the result follows directly by definition.

Comment
This result is applied in various proofs (for example Wilson's Theorem) as:
 * $$-1 \equiv p - 1 \pmod p$$

where $$p$$ is a prime number.