If $a\equiv b \pmod{n}$ and the integers $a, b$, and $n$ are divisible by $d>0$ then $\frac{a}{d}\equiv\frac{b}{d} \pmod{\frac{n}{d}}$
$ac\equiv bc \pmod{\!m}\!\!\iff\!\! a\equiv b \pmod{\!\!\frac{m}{(c,m)}}$ [Euclid's Lemma, congruence form]