If $a|b$ and $c|d$ then $ac|bd$?

Question

If $a|b$ and $c|d$ then $ac|bd$.

Now $aq=b$ and $cr=d$. So multiply both equations we get result. Is this correct ?

Answer 1

I think you mean

If $a\mid b$ and $c\mid d$ then $ac\mid bd$.

And yes your proof is correct.

Alternatively, we have that $$\frac{bd}{ac}=\frac ba\cdot\frac dc$$ which is an integer since it is the product of two integers.