Question

If $c>0$, prove that $\left(ca,cb\right)=c\left(a,b\right)$.

Short answer

It is proved that if $c>0$ then $\left(ca,cb\right)=c\left(a,b\right)$.

Step-by-step solution

Theorem

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

Check for part  

Assume that$\left(a,b\right)=d$ then there exist some integers $x,y$, such that, $ax+by=d$.

As $cd$ divides $ca$ and $cb$, then $cd|\left(ca,cb\right)$.

Then by using $ax+by=d$, we will get:

$cd=cax+cby$

Which implies that:

$\left(ca,cb\right)|cd$

As all these quantities are positive, we will get:

$cd=\left(ca,cb\right)$

Substitute $\left(a,b\right)=d$ into $cd=\left(ca,cb\right)$.

$c\left(a,b\right)=\left(ca,cb\right)$

So, if $c>0$, then $c\left(a,b\right)=\left(ca,cb\right)$.