Let be $G=\langle x\rangle$ a cyclic group and $H$ the subgroup of $G$ that contains $x^m$ and $x^n$. I have to prove that $H=\langle x^{(n,m)}\rangle$; where $(n,m)$ denotes the greatest common divisor of $m$ and $n$.
I know that there exist $\alpha,\beta\in\mathbb{Z}$ such that $$\alpha n + \beta m = (n,m);$$ which implies, since $H\leq G$ and $x^m,x^n\in H$, that $$x^{(n,m)} = x^{\alpha n + \beta m} = x^{\alpha n}x^{\beta m}\in H.$$ This proves that $\langle x^{(n,m)}\rangle\subseteq H$.
I'm having a hard time trying to prove that $H\subseteq \langle x^{(m,n)}\rangle$. I'd appreciate any help.