The structure of $(\mathbb Z/525\mathbb Z)^\times$

The structure of $(\mathbb Z/525\mathbb Z)^\times$

I am working on the following problem.
Find the number of the elements of order 4 in $(\mathbb Z/525\mathbb
Z)^\times$.
I tried to solve it in the following way: since $525=3\cdot5^2\cdot7$, we
have by the CRT $$ (\mathbb Z/525\mathbb Z)^\times \cong (\mathbb
Z/3\mathbb Z)^\times \times (\mathbb Z/25\mathbb Z)^\times \times (\mathbb
Z/7\mathbb Z)^\times. $$
By the fact stated by a Wikipedia article, the consituent groups on the
RHS are isomorphic to the cyclic groups of order 2, 20, 6, respectively.
Thus, the order of an element in $(\mathbb Z/525\mathbb Z)^\times$ is the
least common multiple of some subset $S\subset\{2, 20, 6\}$, which can
never be 4. In conclusion, the number of the elements of order 4 in
$(\mathbb Z/525\mathbb Z)^\times$ is 0.
My question is whether my reasoning is correct (which I doubt because the
result is so trivial). I would also like to ask where accessible proofs of
the fact on Wikipedia (preferably on the Web) can be found.
I would appreciate your help.