I am struggling to prove the following statement: Let $p, q$ be coprime numbers. For all $n \in \mathbb{N}^*$, we define $t_n$ as the order of $[q]$ in $(\mathbb{Z}/{p^n}\mathbb{Z})^\ast$. Show that $(t_{n+1}/t_{n})$ is stationary.
I managed to reduce the problem: let $p = \Pi_{i=1}^m p_i ^{\alpha_i}$ (prime factors decomposition). By the Chinese remainder theorem, the ring isomorphism induces a group isomorphism: $(\mathbb{Z}/p^n\mathbb{Z})^\ast = (\mathbb{Z}/\Pi_{i=1}^m p_i ^{n\alpha_i}\mathbb{Z})^\ast \cong \Pi_{i=1}^m (\mathbb{Z}/p_i ^{n\alpha_i}\mathbb{Z})^\ast $. From there, we can observe that $t_n$ is the lcm of the orders in the $(\mathbb{Z}/p_i ^{n\alpha_i}\mathbb{Z})^\ast$, and the stationnarity of the sequences in $(\mathbb{Z}/p_i ^{n\alpha_i}\mathbb{Z})^\ast$ implies the stationnarity of $(t_n)$, which, by the way, is non-null.
So let's work only with $p$ prime. I was hoping the sequence of orders in $(\mathbb{Z}/p ^{n}\mathbb{Z})^\ast$ to be decreasing, which is not true (try $q = 2, p = 3$). Starting there I am struggling. I only know that $t_n \mid$ card$(\mathbb{Z}/p ^{n}\mathbb{Z})^\ast = \phi(p^n) = p^n - p^{n-1}$( by Lagrange's theorem).
How should I proceed?
Thank you for your help.