Chapter 4.4, Problem 18E

a) Generalize the result in part (a) of Exercise 196; that is, show that if p is a prime, the positive integers less than p, except 1 and $p-\text{1}$ , can be split into $\left(p-\text{3}\right)/\text{2}$ pairs of integers such that each pair consist of integers that are inverses of each other [Hint: Use the result of Exercise 17.]

b) From part (a) conclude that $\left(p-\text{1}\right)!\equiv -\text{1}\left(\text{mod }p\right)$ whenever p is prime. This result is known as Wilson’s theorem.

c) What can we conclude if n is a positive integer such that $\left(n-\text{1}\right)!\ne -\text{1}\left(\text{mod }n\right)$ ?