Final answer:
There is no integer z such that x² + y² = z² when x and y are both odd because the sum of the squares of two odd numbers is even but not divisible by 4, which contradicts the possibility of z being an integer that squares to an even number divisible by 4.
Step-by-step explanation:
The question is asking to prove that there is no integer z such that x² + y² = z² when x and y are both odd integers. This statement is related to the Pythagorean theorem, which involves integers known as Pythagorean triples. However, when x and y are odd, the square of an odd number is also odd (odd² = odd), and the sum of two odd numbers is even (odd + odd = even). So, x² + y² would be even. Thus, z² also has to be even as squares of integers are non-negative. Since the squares of even numbers are divisible by 4 but x² + y² is only divisible by 2 (not by 4), it cannot be equal to the square of an integer z. Therefore, there is no integer z that can satisfy the equation when both x and y are odd.