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For each of the following statements, indicate whether the statement is true or false and justify your answer with a proof or a counterexample. (a) Let m ∈ N and let a, b, c ∈ Z. If m ∤ c and ca ≡ cb (mod m), then a ≡ b (mod m). (b) If m ∈ N, then the only solutions of the equation X² = X in Z/mZ are 0Z/mZ and 1Z/mZ. (c) Let m ∈ N with m > 2, and let α ∈ Z/mZ be nonzero. If the equation X² = α admits a solution in Z/mZ, then it admits exactly two solutions in Z/mZ.
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