1) Define H = {0, 3, 6, 9} ⊂ Z12. Let R be a relation on Z12 defined by the following: aRb ⇔ a ⊖ b ∈ H.
(a) Prove that H is a group under modular addition. (b) Prove R is an equivalence relation on Z12.
(c) Give a full description of the equivalence classes given by R.

2) Let p, q > 1 be consecutive odd numbers.
(a) Show(p+q)² =4pq+4.
(b) Show(p+q)² ≡4 modpq.
(c) Show that 4−1 exists in Zpq.
(d) Show that the set
{x ∈ Z × pq|x = x⁻¹}
has at least three distinct elements in it. Hint: 1 and pq − 1 are two of them.

3) Let a be a positive integer. Prove that the sum of a consecutive integers is divisible by a if and only if a is odd.

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