Let n > 1 be an integer. Prove that ℤ /n has exactly n elements. Hint 1: Use the following strategy: 1) Prove that ℤ /n = {[0], [1], · · · , [n − 1]}. (This shows that |ℤ /n| ≤ n.) 2) Prove that [i] 6= [j] whenever 0 ≤ i < j ≤ n − 1. (This shows that [0], · · · , [n − 1] are all distinct, hence |ℤ /n| ≥ n.)

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