Question

The statement $P(n) = 9^{n} - 8^{n}$, when divided by 8, always leaves the remainder

• A. 1
• B. 2
• C. 3
• D. 7

Hint:

To find remainders of expressions like $(8k+1)^n$, use the binomial theorem — all terms except the constant 1 will be divisible by 8.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Use the binomial expansion to express $9^n$ in terms of multiples of 8.
Step 2: Detailed Explanation:
$9^n = (8+1)^n = \displaystyle\sum_{k=0}^n \binom{n}{k}8^k = 1 + 8n + \binom{n}{2}8^2 + \cdots = 1 + 8(\text{integer})$.
So $9^n \equiv 1 \pmod{8}$, which gives $9^n - 8^n \equiv 1 - 0 = 1 \pmod{8}$.
Step 3: Final Answer:
The remainder is always $1$.