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(9 marks) Define the relation τ on Z by a τ b if and only if...

Define the relation τ on Z by a τ b if and only if there exists x ∈ {1, 4, 16} such that ax ≡ b (mod 63).

(a) Prove that τ is an equivalence relation.

(b) Prove that there exists an integer n with 1 ≤ n ≤ 62 such that the equivalence class of n is {m ∈ Z | m ≡ n (mod 63)}.

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