Is the following statement true and why/why not?

Question

Assume $1 \le n$. And $x \in Z_n = \{0,1,2,..n-1\}$. Assume in addition that for some $k>0$,

$kx = 0 \pmod n$, i.e. $\,n\mid kx$. And assume $k\mid n$.

Is it then true that $\gcd(n,x)=n/k $, where $\gcd=$ greatest common divisor? Note by the assumptions, $n/k$ must divide both $n$ and $x$, but how can I show that this is in fact the greatest common divisor? Thanks,

Answer 1

If you suppose $gcd(n,x)=1$, as another answer states it, it is clear.

If you don't, then the result is false : take $n=16$, $k=4$ and $x=8$.

Answer 2

If $n|kx$ and $gcd(n,x)=1$, then $n|k$. Adding the assumption that $k|n$, you get that $n=k$.

So indeed $1=gcd(n,x)=n/k$, but it is a strange way of writing it since $k=n$ !