Pythagoras in trouble

Suppose x, y and z are co-prime positive integers satisfying the Pythagoras theorem, x^2 +y^2 = z^2

Now, what can we say about the nature of x and y?

  1. Both x and y are even
  2. Both x and y are odd
  3. x and y are either both odd or both even
  4. Only one of x and y can be even,
  5. There is no such constraint, x and y can be any numbers at a time

Put the correct option number in the answer. Please share your solutions, because here, the method, rather than the answer is the most important thing.

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