Prove that if a|b, c|d, then ac|bd

Question

I'm trying to prove it, but I can't find how.

If a divides b, and c divides d, then a*c divides b*d

Answer 1

Hint: $a\mid b$ means that there exists an integer $k$ such that $b = ka$.

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You seem to have written the essential step as a comment. Here's how it would fit into a complete proof:

Suppose that $a\mid b$ and $c \mid d$. It follows that we have $b = k_1 a$ and $d = k_2 c$ for integers $k_1, k_2$. It follows that $$ bd = (k_1a)(k_2c) = (k_1k_2)(ac) $$ Let $k$ be equal to the integer $k_1 k_2$. We see that $(bd) = k(ac)$. Thus, $bd$ is divisible by $ac$.

Answer 2

It goes directly from the definition of divisibility. If $a|b$ then exists $k$ such that $b=ka$. If $c|d$ then exists $l$ such that $d=lc$. Hence $bd=klac$.