GCD from Congruence Modulo m

From ProofWiki

Jump to: navigation, search

Theorem

Let $a, b \in \Z, m \in \N$.

Let $a$ be congruent to $b$ modulo $m$.

Then the GCD of $a$ and $m$ is equal to the GCD of $b$ and $m$.

That is:

$a \equiv b \pmod m \implies \gcd \left\{{a, m}\right\} = \gcd \left\{{b, m}\right\}$

We have:

$a \equiv b \pmod m \implies \exists k \in \Z: a = b + k m$

Thus:

$a = b + k m$

and the result follows directly from GCD with Remainder.

$\blacksquare$