2022 · Q29
Consider the question and two statements given below : Question : Is (xⁿ + yⁿ) divisible by (x + y)?
Statement 1n is a natural number.
Statement 2n is an even natural numberWhich one of the following is correct in respect of the question and the statements?
Elementary MathematicsNumber System
- AStatement-1 alone is sufficient to answer the question
- BStatement-2 alone is sufficient to answer the question
- CBoth Statement-1 and Statement-2 are sufficient to answer the question
- DBoth Statement-1 and Statement-2 are not sufficient to answer the question
Explanation
By the factor theorem, xⁿ+yⁿ is divisible by x+y exactly when n is odd (for even n it is not, in the generic case). Statement 1 leaves the parity of n open, so the question cannot be answered. Statement 2 (n even) settles it in the negative, which is a definite answer. Hence statement 2 alone is sufficient. The keyed option is B.