All variable are $\in \mathbb{Z}$. $k$ divides $n$ and $k$ divides $m$ therefore because $k$ is a divisor of $n$ and $m$ it must be also a divisor for the greatest common divisor of $n$ and $m$. Assume there is no number $w\in \mathbb{Z}$ such that $kw$ = gcd$(n,m):=z$ then there would be a contradiction since there exists $n',n'',m',m''$ such that $n+m = k(n'+m') = z(n''+m'') \Rightarrow \frac{k(n'+m')}{k(n'+m')} = \frac{z(n''+m'')}{k(n'+m')} = 1 $
That's my thought process so far, Maybe it's the wrong Approach- Maybe I am in the Right direction if the latter is the case please tell me how I should continue otherwise it would be Kind if you could give me the proof.
