Let $a, b$ and $m$ be integers, and $m \ge 2, \ ab ≡ [ (a \bmod m) * (b \bmod m) ] (\bmod m).$

Question

I'm relatively new to this, I got no idea how to proceed at all. I'm just bad at this chapter :( $$ m\mid (ab- (a \bmod m \cdot b \bmod m)). $$ How do I proceed after that? Is this even the first step?

Answer 1

Since, by definition, $m$ divides $a-a\bmod m$ and $b-b\bmod m$, rewrite the expression as \begin{align} ab-a\bmod m\cdot b\bmod m&=(a-a\bmod m)b+a\bmod m\cdot b-a\bmod m\cdot b\bmod m\\ &=(a-a\bmod m)b+a\bmod m\cdot(b-b\bmod m) \end{align}