Prove that if a ≡ b mod m and c ≡ d mod m, then ac ≡ bd mod m. Prove that if a ≡ b mod m and c ≡ d mod m, then ac ≡ bd mod m. 
In previous attempts I have tried to express a as b + mk and c as d + ml and I have also shown that m|a-b and m|c-d but I was unable to reach a complete proof.
Just a problem I have been trying to solve out of interest.

By hypothesis, $a=b+km$ and $c=d+lm$. Then 
$ac = (b+km)(d+lm) = bd + blm + dkm + klm^2 = bd + m(bl+dk+klm)$
and so 
$ac\equiv bd\mod m$. 
