I am trying exercises of Tom Apostol Modular functions and Dirichlet Series in Number Theory and I cannot think about this problem of Chapter 2 .
Problem 2 - For each prime p the number of solutions, distinct mod $p^r$ , of all possible congruences of the form ax-by$\equiv$1 ( mod $ p^r $ ) , where ( a, b, p) =1 is equal to f($p^r$) , where f(n) denotes the number of equivalence classes of matrix modulo n . Index of $\Gamma^n$ in $\Gamma$ is the number of equivalence classes of matrices modulo n.
Before this question there is this question in the book which I proved proved might be useful --> If a, b, n are integers with n$\ge$ 1 and (a, b, n) =1 the congruence ax-by$\equiv$ 1 ( mod n) has exactly n solutions, distinct modulo n. But I am not able to see how to use the concept of equivalent classes here. Can someone please help