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Define the relation τ on Z by aτ b if and only if there exists x...

  1. Define the relation τ on Z by aτ b if and only if there exists x ∈ {1,4,16} such that

    ax ≡ b (mod 63).

    1. (a) Prove that τ is an equivalence relation.

    2. (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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