Question: What is the remainder when x^{2n} - y^{2n} + 1 is divided by x^n + y^n, where n is a natural number? Statement I: n is odd. Statement II : n is even.
A Question is given followed by two Statements I and II. Consider the Question and the Statements. Which one of the following is correct in respect of the above Question and the Statements?
- A. The Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone
- B. The Question can be answered by using either Statement alone
- C. The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone
- D. The Question can be answered even without using any of the Statements ✓
Correct Answer: D. The Question can be answered even without using any of the Statements
Explanation
The expression x^{2n} - y^{2n} is a difference of squares and factors to (x^n - y^n)(x^n + y^n). Thus, it is always exactly divisible by x^n + y^n. The remainder of (x^{2n} - y^{2n} + 1) divided by x^n + y^n will always be 1, regardless of whether n is odd or even.
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