UPSC CDS II 2024 Math Paper

Concept:If $n$ is odd, $a^n + b^n$ is divisible by $a + b$. Here $a = 327$, $b = 173$, and $a+b = 500$.Explanation:For Statement-I: $n$ is an odd natural number.Then $327^n + 173^n$ is divisible by $327 + 173 = 500$.This is a standard algebraic identity for odd powers.So Statement-I alone is sufficient to answer the question.For Statement-II: $n$ is a positive integer.This includes both odd and even values of $n$.If $n$ is even, the identity does not hold.Example: $n = 2$ gives $327^2 + 173^2 = (327+173)^2 - 2 \times 327 \times 173 = 500^2 - 113142$, which is not a multiple of $500$.Thus, Statement-II alone cannot guarantee divisibility by $500$.Therefore, the question can be answered using Statement-I alone, but not using Statement-II alone.Answer:Option A: The Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone.