Divisibility Property

Question

I am trying to justify the following result:

Let $p,q$ be integers such that $GCD(p,q) = 1$. Then for all $n \in \mathbb{N}$ exists an integer $j_n$ such that $q^{j_n}t = t \ (mod \ p^{2n+1}), \ \forall 0 \leq t < p^{2n+1}$.

Any ideas on how to prove this result in a simple way?

Thanks in advance

Answer 1

Hint: It is sufficient that $q^{j_n} \equiv 1 \mod p^{2n+1}$. Consider the sequence of powers $q$, $q^2$, $q^3$, $\ldots$ modulo $p^{2n+1}$. What happens if $1$ does not occur in this sequence?