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Problem 63 ($|a^k| = |a|/\gcd(k,|a|)$)
Let $a$ be an element of order $n$.
- If $d$ is a divisor of $n$, then prove that $|a^d|=n/d$.
- For any $k\in \mathbb{N}$, prove that $|a^k| = n/\gcd(k,n)$.
You'll need to use the definition of order, and two previous problems to make short work of this problem. In particular, if you pay attention to the fact that the order of an element always matches the order of the subgroup generated by that element, then you can make sense of $|a^k|$ in terms of a greatest common divisor.
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