Question

If $a|c$ and $b|c$, must $ab$ divide ? Justify your answer.

Short answer

N, if $a|c$ and $b|c$, then $ab$must divide$c$ is not true.

Step-by-step solution

Apply definition of divisibility

As $a|c$ and $b|c$, then by definition of divisibility $c=ax$ and $c=by$ for some integer $x$ and $y$.

This implies that:

$ax=by$

Therefore, we can say that $ab$ does not divide $c$.

Consider example 

For example, we know that $4|12$ and $6|12$. Multiply 4 by 6 as:

$4·6=24$

But 24 is not divided by 12.

Hence, if $a|c$ and $b|c$ then $ab$does not divide $c$.