Scaling preserves Modulo Addition
Jump to navigation
Jump to search
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:
- $c x \equiv c y \pmod m$
$\blacksquare$