Let $x*y= \gcd(x,y)$ on $\mathbb N$. I have to prove that $(\mathbb{N} , *)$ is a commutative monoid.
I knew how to show the associativity, but I have a problem on the identity element.
$\gcd(x,e)=x \to x\mid e$ so $e=xk,k \in \mathbb N ^*$
And I don't know what to do next.