Question

If \( 49^n + 16n + P \) is divisible by 64 for all \( n \in \mathbb{N} \), then the least negative integral value of \( P \) is:

• A. -2
• B. 3
• C. -4
• D. -1

Hint:

When solving modular arithmetic problems, try to reduce the given numbers modulo the specified value, and check for patterns.

Answer 1

The Correct Option is D

We are given that \( 49^n + 16n + P \) must be divisible by 64 for all \( n \in \mathbb{N} \). Step 1: Simplify the equation First, rewrite the given expression: \[ 49^n + 16n + P \equiv 0 \pmod{64} \] Note that \( 49 \equiv 49 \pmod{64} \), so we want to find the behavior of \( 49^n \mod 64 \). Step 2: Calculate \( 49^n \mod 64 \) Since \( 49^2 \equiv 49 \pmod{64} \), we find that \( 49^n \equiv 49 \pmod{64} \) for any \( n \geq 1 \). Thus, the expression becomes: \[ 49 + 16n + P \equiv 0 \pmod{64} \] Step 3: Solve for \( P \) For \( n = 0 \): \[ 49 + P \equiv 0 \pmod{64} \] \[ P \equiv -49 \pmod{64} \] Since \( -49 \equiv 15 \pmod{64} \), the least negative value for \( P \) is \( -1 \). Thus, the correct answer is \( -1 \).