Question

If \( 4^n + 15n + P \) is divisible by 9 for all \( n \in \mathbb{N} \), then the least negative integral value of \( P \) is

• A. -1
• B. -2
• C. -3
• D. -4

Hint:

Use modular arithmetic and plug in small values to find patterns for divisibility.

Answer 1

The Correct Option is A

We need \( 4^n + 15n + P \equiv 0 \pmod{9} \) for all \( n \in \mathbb{N} \)
Let’s try small values of \( n \) and check modular values.
Step 1: Use \( n = 1 \)
\( 4^1 + 15 \cdot 1 + P = 4 + 15 + P = 19 + P \equiv 0 \pmod{9} \Rightarrow P \equiv -1 \pmod{9} \)
Step 2: Try \( P = -1 \)
Try \( n = 2 \): \( 4^2 + 15 \cdot 2 -1 = 16 + 30 - 1 = 45 \equiv 0 \pmod{9} \)
Try \( n = 3 \): \( 4^3 + 15 \cdot 3 - 1 = 64 + 45 - 1 = 108 \equiv 0 \pmod{9} \)
So \( P = -1 \) satisfies all cases.