Congruence Modulo Negative 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.