If $n,m$ are positive integers such that $m \mid n$, then the natural map $$(\mathbb{Z}/n\mathbb{Z})^{\times} \to (\mathbb{Z}/m\mathbb{Z})^{\times},~ [z] \mapsto [z]$$ is surjective. This has appeared a number of times on mathoverflow and math.stackexchange. See also MO/31495 for more general observations.
But I am not asking for a proof. I wonder if someone knows a reference in the literature for this fact. I would like to cite it. So a blog post or lecture notes unfortunately are not good enough. Ideally it should be a textbook, ideally written in english.
An equivalent statement is the following (and actually this formulation is what I need): The automorphism group of $\mathbb{Z}/n\mathbb{Z}$ acts transitively on the subset of elements of order $m$.
