Law of Inverses (Modulo Arithmetic)/Corollary 1

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 $19$