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Intersection of cyclic subgroups: $(x^m) \cap (x^n) = (x^{lcm(m,n)})$

If $G=\langle x\rangle$ is a finite cyclic group that $\langle x^n\rangle \cap \langle x^m\rangle = \langle x^{lcm(m,n)}\rangle$ for all integers $m$ and $n$, where $\langle x\rangle$ is the group generated by $x$. One of those easy questions from undergrad I'm just not seeing.

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