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In a solution to some problem, the writer uses the following claim:

If $m$ divides $n$ then a cyclic group of order $n$ contains $(\mathbb{Z}/m\mathbb{Z})^\times$ elements of order $m$.

Is there some known theorem that this is based on? (ideally a reference to a relevant theorem or two from a book). I've tried looking at Hestein's Topics in Algebra but nothing there seems relevant.

Anon
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  • As an exercise, prove that if $g$ has order $m$ then the order of $g^k$ is $\frac m{\gcd(m,k)}$ – lulu Feb 25 '23 at 15:38
  • You mean $|(\Bbb Z/m\Bbb Z)^{\times})|=\phi(m)$ many elements. This is proved in the duplicate, with $d=m$ a divisor of $n$. – Dietrich Burde Feb 25 '23 at 15:44

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