Prove if $a \equiv c \pmod n$ and $b \equiv d \pmod n$ then $ab \equiv cd \pmod n$. Prove if $a \equiv c \pmod{n}$ and $b \equiv d \pmod n$ then $ab \equiv cd \pmod{n}$.
I tried to use $(a-c)(b-d) = ab-ad-cb+cd$, but it seem doesn't work.

You need to subtract and add the same thing:
$$ab-cd=ab-ad+ad-cd=\ldots$$
