Question

If $a|b$ and $c|d$, prove that $ac|bd$.

Short answer

It is proved that if $a|b$ and $c|d$ then $ac|bd$.

Step-by-step solution

Apply divisibility definition 

As $a|b$ then by definition of divisibility, we have $b=ax$ for some integer $x$ and $a\ne x$.

As $c|d$ then by definition of divisibility, we have $d=cy$ for some integer $y$ and $b\ne 0$.

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

Determine the value of $bs$ as:

$\begin{array}{rcl}bd& =& \left(ax\right)\left(cy\right)\\ & =& \left(ac\right)\left(xy\right)\end{array}$

This implies that role="math" localid="1646125167381" $ac|bd$ for some integer $xy$.

Hence, it is proved that if $a|b$ and $c|d$ then $ac|bd$.