2025 · Q81
A Question is given followed by two Statements I and II.
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 In is odd.
Statement IIn is even.
Which one of the following is correct?
Elementary MathematicsAlgebra
- AThe Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone
- BThe Question can be answered by using either Statement alone
- CThe Question can be answered by using both the Statements together, but cannot be answered using either Statement alone
- DThe Question can be answered even without using any of the Statements
Explanation
x^(2n)-y^(2n)=(x^n-y^n)(x^n+y^n), so x^(2n)-y^(2n)+1=(x^n+y^n)(x^n-y^n)+1 with remainder 1 for every natural n. No statement is needed. Testing parities separately wastes time.