x and y are integers, 0 < x < y, and x² + y^2 is even. Which of the following integers must be even?

Rationale

x + y must be even.

For the sum of two integers x and y to produce an even result, both integers must either be even or odd. Given that x² + y² is even, it implies that both x and y are either even or odd.

A (xy) YOUR ANSWER

The product xy can be even or odd depending on the parity of x and y. If both x and y are even, xy is even. However, if both are odd, xy is odd. Thus, xy does not necessarily have to be even.

B (x + y) CORRECT

Since x² + y² is even, both integers x and y must share the same parity (both odd or both even). Therefore, their sum x + y must also be even, confirming that this option is correct.

C (y - x) YOUR ANSWER

The difference y - x can also be either even or odd. If both x and y are even, y - x is even. If both are odd, y - x remains even. However, if one were even and the other odd, y - x would be odd. This inconsistency means y - x is not guaranteed to be even.

D (x² + y) YOUR ANSWER

The expression x² + y can yield either an even or odd result. If x is even, x² is even, and if y is odd, the sum becomes odd. Conversely, if x is odd and y is even, then x² is odd and the sum also becomes odd. Thus, x² + y does not necessarily have to be even.

Conclusion

The requirement that x² + y² is even indicates that both integers x and y must be of the same parity, ensuring that their sum x + y is even. The other expressions can vary in parity and do not guarantee an even result, making x + y the only universally valid answer. This understanding is crucial in problems requiring the determination of integer properties based on their parities.

x and y are integers, 0 < x < y, and x² + y^2 is even. Which of the following integers must be even?

x + y must be even.

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