Proof that a ≡ b (mod m) and c ≡ d (mod m) implies a + c ≡ b + d (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:
$$\begin{gathered} a + c = b + d + mk + ml \\ a + c = b + d + m(k + l) \end{gathered}$$