Suppose $G$ be a finite group, $a \in G$ has finite order, then check whether the formula true or false? $ \langle a^m\rangle \cap\langle a^n\rangle=\langle a^{lcm(m,n)mod|a|}\rangle$
My attempt to prove the result,
Proof. Let $x \in \langle a^{lcm(m,n)mod|a|}\rangle.$
We know that $m|lcm(m,n)$ and $n|lcm(m,n).$
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How to to proceed further to obtain $x \in \langle a^m\rangle$ and $x \in\langle a^n\rangle$
Conversely,
$x \in \langle a^m\rangle\cap\langle a^n\rangle$
$x \in \langle a^m\rangle$ and $x \in\langle a^n\rangle $
$x=a^{ms}$ and $x=a^{nt},$ where $s,t \in \mathbb{Z} $
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How to proceed further to prove $x \in \langle a^{lcm(m,n)mod|a|}\rangle$
Please provide hints to complete the proof. Is this formula false? I am not able to find the counter examples. If this formula false, what are the extra conditions required to make it correct?