Proof that a ≡ b (mod m) and c ≡ d (mod m) implies ac ≡ bd (mod m)

Modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. If $a ≡ b \bmod m$ and $c ≡ d \bmod m$:

$$\begin{gathered} a = b + mk \\ c = d + ml\end{gathered}$$

If we add them:

$$\begin{gathered} ac = (b + mk)(d + ml) \\ ac = bd + bml + dmk + mkml = bd + m(b+d+mkl) \end{gathered}$$

This proves that $ac ≡ bd \bmod m$. As a corollary, we can also claim $a^k ≡ b^k \bmod m$.