If x and y are natural numbers such that x + y = 2017, what is the exact value of ( -1 )^x + ( -1 )^y?

Introduction / Context:
This question tests parity (odd/even) and how the expression ( -1 )^n behaves based on whether n is even or odd. The key idea is that powers of -1 alternate between 1 and -1, so the sum ( -1 )^x + ( -1 )^y depends only on whether x and y are even or odd.

Given Data / Assumptions:

• x and y are natural numbers.
• x + y = 2017 (an odd number).

Concept / Approach:
Use the fact that ( -1 )^even = 1 and ( -1 )^odd = -1. Also, if a sum is odd, then one number must be even and the other must be odd (they cannot both be even or both be odd).

Step-by-Step Solution:

Verification / Alternative check:
Example check: take x = 100 (even) and y = 1917 (odd). Then ( -1 )^100 = 1 and ( -1 )^1917 = -1, sum = 0. Any other even-odd pair gives the same result.

Why Other Options Are Wrong:

Common Pitfalls:
Forgetting that an odd sum forces opposite parity, or assuming x and y can both be odd because 2017 is odd (actually odd + odd is even).

Final Answer:
0