Question 10

x + y must be even.

Since x and y are integers and the sum x² + y² is even, both x and y must be either both even or both odd. This implies that their sum, x + y, will also be even.

A) xy

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

B) x + y

The sum x + y is indeed even. Given that both x and y must be either even or odd for x² + y² to be even, their sum will always yield an even result regardless of their individual values.

C) y - x

The difference y - x can also be either even or odd. If both x and y are odd, y - x will be even; however, if both are even, y - x will still be even. Thus, y - x does not have to be consistently even across all scenarios.

D) x² + y

The expression x² + y can be even or odd depending on the parity of x and y. If x is odd and y is odd, then x² is odd, making x² + y odd. Conversely, if both are even, then x² + y is even. Therefore, this option does not guarantee an even result.

Conclusion

The only option that must be even is x + y, as it is the direct result of the parity conditions established by the problem. All other options can vary between even and odd based on the specific values of x and y, making them unreliable as definitive answers.