Integer is Divisor Modulo m of Every Integer iff Coprime to m

Theorem

Let $m, n \in \Z$ be integers.

Then:

$n$ is coprime to $m$

if and only if:

$n$ is a divisor modulo $m$ to every integer.

Proof

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Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): factor modulo $n$
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): factor modulo $n$