Law of Inverses (Modulo Arithmetic)/Corollary 1
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Corollary to Law of Inverses (Modulo Arithmetic)
Let $m, n \in \Z$ such that:
- $m \perp n$
that is, such that $m$ and $n$ are coprime.
Then:
- $\exists n' \in \Z: n n' \equiv 1 \pmod m$
Proof
By definition of coprime:
- $m \perp n \iff \gcd \set {m, n} = 1$
The result follows directly from Law of Inverses (Modulo Arithmetic).
$\blacksquare$
Sources
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.4$: Integer Functions and Elementary Number Theory: Exercise $19$