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.