Question 66
Given an odd number $n \gt 1$, let $S = \{k : 1 \le k \lt n, (k,n) = 1\}$ and let $T = \{k : k \in S, (k+1,n) = 1\}$. For each $k \in S$, let $r_k$ be the remainder left by $(k^{|S|} - 1)/n$ upon division by $n$. Show that
$$
\prod_{k \in T} (r_k - r_{n-k}) \equiv |S|^{|T|} \pmod{n}.
$$
$$
\prod_{k \in T} (r_k - r_{n-k}) \equiv |S|^{|T|} \pmod{n}.
$$