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For any natural number n, the expression x^(2n - 1) + y^(2n - 1) is divisible by which of the following?
- x + y
- x - y
- x² + y²
- x² + xy
Correct answer: x + y
Solution
For any natural n, x^(2n-1) + y^(2n-1) has an odd exponent, and a^m + b^m is divisible by a + b whenever m is odd. Hence x^(2n-1) + y^(2n-1) is divisible by x + y (e.g. x^3 + y^3 = (x+y)(x^2 - xy + y^2)).
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