Negative Number is Congruent to Modulus minus Number
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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}$