Prove if a | b, then a | bc for all integers c ,true

Question

Please prove that ) if a | b, then a | bc for all integers c;

my solution: b= a x j c= a x d and I don't know what do I have to do next or how can I have a good proof.

Answer 1

If $a\mid b$ then $b=ak$ for $k\in\mathbb{Z}$. We want to show that $a\mid bc$, i.e. we need to show that there exists some integer $m$ where $bc=am$.

$$bc=(ak)c=a(kc)$$ so we have found an $m$ that works, namely $m=kc$ so $a\mid bc$.