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 $a \equiv b mod m$ and $c \equiv d mod m$ :

\begin{matrix} a = b + m k \\ c = d + m l \end{matrix}

If we add them:

\begin{matrix} a c = (b + m k) (d + m l) \\ a c = b d + b m l + d m k + m k m l = b d + m (b + d + m k l) \end{matrix}

This proves that $a c \equiv b d mod m$ . As a corollary, we can also claim $a^{k} \equiv b^{k} mod m$ .

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