Question: If p is a positive integer, then what is the remainder when p^{n} is divided by p+1?<br>Statement-I: n is even.<br>Statement-II: p is even.

Consider the following for the next ten (10) items that follow : Each item contains a Question followed by two Statements. Answer each item using the following instructions : Choose option (a) If the Question can be answered by one of the Statements alone, but not by the other. (b) If the Question can be answered by either Statement alone. (c) If the Question can be answered by using both the Statements together, but cannot be answered by using either Statement alone. (d) If the Question cannot be answered even by using both Statements together.

  1. A. If the Question can be answered by one of the Statements alone, but not by the other. ✓
  2. B. If the Question can be answered by either Statement alone.
  3. C. If the Question can be answered by using both the Statements together, but cannot be answered by using either Statement alone.
  4. D. If the Question cannot be answered even by using both Statements together.

Correct Answer: A. If the Question can be answered by one of the Statements alone, but not by the other.

Explanation

We need to find the remainder when p^n is divided by p+1. Using the polynomial remainder theorem, we can conceptually substitute p \equiv -1 \pmod{p+1} into the expression, which yields (-1)^n. If n is even, (-1)^n = 1, giving a constant remainder of 1 regardless of the value of p. If n is odd, the remainder would be -1, which is equivalent to p in modulo arithmetic. Therefore, knowing whether n is even or odd (Statement-I) is fully sufficient to determine the exact remainder, while Statement-II is completely irrelevant.

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