identity element of $(D_n ,\gcd)$

Question

Let $x*y=\gcd(x,y)$ on $D_n=\{x\in \mathbb N:x |n\}$. I have to prove that $(D_n,*)$ is a commutative monoid and I have a problem when finding the identity element.

$x*e=x$

So $gcd(x,e)=x \to e=xk,k\in \mathbb N^*$ and $e \leq n$

A good $e$ is $0$ but it's not in $D_n$

Answer 1

If $x \in D_n$, then $x$ divides $n$. Thus $\gcd(n,x) = x$. Thus $n$ is the identity element of your monoid.