CDS 2024 · PYQ · Data Sufficiency / Number Theory · medium
Question: If p is a positive integer, then what is the remainder when p^n is divided by p + 1? Statement-I: n is even. Statement-II: p is even.
A.If the Question can be answered by one of the Statements alone, but not by the other.✓ Correct
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.
Explanation
p ≡ -1 (mod p+1), so p^n ≡ (-1)^n (mod p+1). If n is even, p^n ≡ 1 (mod p+1), so the remainder is 1. Statement I alone determines the remainder. Statement II (p even) tells us p+1 is odd but doesn't fix the parity of n, so the remainder could be 1 or p (i.e., -1 mod p+1). Hence only Statement I alone suffices.
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