prove that $\forall k\in \mathbb Z$ if $(m,n)=1$(coprimes) then $(k,m,n)=1$

Question

Prove that $\forall k\in \mathbb Z$ if $(m,n)=1$(coprimes) then $(k,m,n)=1$

I did this by contradiction:

let $d=(k,m,n)$ so that $d\neq 1$; by definition of greatest common divisor $d|k, d|m, d|n$ and by hypothesis we have that $(m,n)=1$ so that means that $d$ must be equal to 1 but this is a contradiction so $(k,m,n)=1$

I just want you to tell me if this prove is correct please

Answer 1

It is correct, though you may want to expand a bit on why $d$ must be equal to 1 (use the remaining property of the greatest common divisor).

It may also be more elegant without the use of contradiction. Start with $d = (k,m,n)$ without the $d \neq 1$ condition. Following your argument, $d = 1$ so $(k,m,n)=1$.

Answer 2

Yes, the proof is fine. Directly it is: $\,(k,m,n)\mid m,n\,\Rightarrow (k,m,n)\mid (m,n)$

Alternatively: $\,\ (k,m,n) = (k,(m,n))\mid (m,n)$