Please explain why these proofs are incorrect and provide a correct answer: 1/ Prove that if x, y, z ∈ Z and x2 + y2= z2

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Please explain why these proofs are incorrect and provide a correct answer:
1/ Prove that if x, y, z ∈ Z and x2 + y2= z2 , then 3|xy
Proof by Contradiction: Assume x and y are not divisible by 3. So, x and y can be congruent to -1 and 1. Thus, x2 and y2 equal 1. Since, 1+1 = 2. => z = √ 2 (why this answer is incorrect)
2/ Prove that when an irrational number is divided by a nonzero rational number, the resulting number is irrational.
Proof: For the sake of contradiction assume an irrational, x, divided by a rational number, a/b, is rational y.
So (a/b)x = y
So x has to be rational as y is rational.