Question

Suppose $\left(a,b\right)=1$, if $a|b$ and $b|c$, prove that $ab|c$.

Short answer

It is proved that $ab|c$.

Step-by-step solution

Theorem

According to the theorem, if $a|bc$ and $\left(a,b\right)=1$, then $a|c$.

Check for part  ab|c

It is given that, $a|c$ and $b|c$, then $\left(a,b\right)=1$. Such that:

$a|t$

Multiply both sides by $b$.

$ab|bt=c$

Which implies that $ab|c$.