$\langle a^k\rangle = \langle a^{\gcd(k,|a|)}\rangle$

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Problem 62 ($\langle a^k\rangle = \langle a^{\gcd(k,|a|)}\rangle$)

Let $a$ be an element of order $n$ and let $k\in\mathbb{N}$. Prove that $\langle a^k\rangle = \langle a^{\gcd(k,n)}\rangle$.

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