About order of $q \pmod {p^n}.$

Question

How to prove this affirmation:

If $q$ has order $\phi (p^n) \pmod{p^n}$, so $q$ also has order $\phi (p^{n-i+1})\pmod{p^{n-i+1}}$ , $1 \leq i \leq n$, where $\gcd(q,p)=1$ and $\phi$ is the Euler's totient function.

Answer 1

If the different powers of $q$ hit every possible (i.e. coprime to $p$) congruence class modulo $p^n$ (which is what it means that $q$ has order $\phi(p^n)$), then they necessarily have to hit every possible congruence class modulo smaller powers of $p$ as well.