What is the remainder when 9³ + 9⁴ + 9⁵ + 9⁶ + ... + 9¹⁰⁰ is divided by 6?

Question

What is the remainder when 9³ + 9⁴ + 9⁵ + 9⁶ + ... + 9¹⁰⁰ is divided by 6?

1. A. 0 (Correct answer)
2. B. 1
3. C. 2
4. D. 3

Correct Answer

Option A — 0

Detailed Solution & Explanation

The correct answer is 0.

Key Points

• 9 ≡ 3 (mod 6). 9² = 81 ≡ 3 (mod 6).
• In fact 9ⁿ ≡ 3 (mod 6) for all n ≥ 1. Sum = 9³+9⁴+...+9¹⁰⁰ = 98 terms each ≡ 3 (mod 6).
• Sum ≡ 98×3 = 294 ≡ 294 mod 6 = 0 (since 294 = 49×6). Answer: 0.