Question: (a) Find all x in (Z5 ) such that x = x-1. (b) Find all x in (Z11, ) such that x = x-1. (c)

(a) Find all x in (Z*5 ∙) such that x = x-1.
(b) Find all x in (Z*11, ∙) such that x = x-1.
(c) Let p be a prime. Find all x in (Z*p, ∙) such that x = x-1.
(d) Prove that (p - 1)! ≡ - 1 (mod p), for p a prime. [This result is known as Wilson's Theorem, although it was only conjectured by John Wilson (1741-1793). The first proof was given in 1770 by Joseph Louis Lagrange (1736-1813).]

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