Scaling preserves Modulo Addition

Theorem

Let $m \in \Z_{> 0}$.

Let $x, y, c \in \Z$.

Let $x \equiv y \pmod m$.

Then:

$c x \equiv c y \pmod m$

Proof

Let $x \equiv y \pmod m$.

Then by definition of congruence:

$\exists k \in Z: x - y = k m$

Hence:

$c x - c y = c k m$

and so by definition of congruence:

$c x \equiv c y \pmod m$

$\blacksquare$