If \(a ≡ b mod m\), then there is an integer \(k\) such that:
$$ mk = b - a $$
We can multiply both sides with integer \(c\):
$$ \begin{gathered} c(mk) = c(b - a) = cb - ca \\ \implies ca ≡ cb \bmod cm \end{gathered}$$
If \(a ≡ b mod m\), then there is an integer \(k\) such that:
We can multiply both sides with integer \(c\):