Question

If $a|c$ and $b|c$, prove that$a|c$ .

Short answer

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

Step-by-step solution

Apply divisibility definition 

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

Prove that if a|b  and  b|c then  a|c

Substitute $b=ax$ into$c=by$ and simplify as:

$\begin{array}{rcl}c& =& by\\ & =& \left(ax\right)y\\ & =& a\left(xy\right)\end{array}$

This implies that $c|a$ for some integer$xy$ .

Thus, it is proved that if role="math" localid="1646119111193" $a|b$ and $b|c$ then $a|c$.