Question 88
Given an odd number $n \ge 3$, let
$$
S = \{k : 1 \le k \le n, \gcd(k, n) = 1\}
$$
and let
$$
T = \{k : k \in S, \gcd(k + 1, n) = 1\}.
$$
For each $k \in S$, let $r_k$ be the remainder left by $\frac{k|S| - 1}{n}$ upon division by $n$. Show that
$$
\prod_{k \in T} (r_k - r_{n-k}) \equiv |S|^{|T|} \pmod{n}.
$$
$$
S = \{k : 1 \le k \le n, \gcd(k, n) = 1\}
$$
and let
$$
T = \{k : k \in S, \gcd(k + 1, n) = 1\}.
$$
For each $k \in S$, let $r_k$ be the remainder left by $\frac{k|S| - 1}{n}$ upon division by $n$. Show that
$$
\prod_{k \in T} (r_k - r_{n-k}) \equiv |S|^{|T|} \pmod{n}.
$$