Correct Option (1)

To determine the remainder when the expression 9³ + 9⁴ + ... + 9¹⁰⁰ is divided by 6, we apply principles of modular arithmetic.

Analysis of Individual Terms:
- First, consider the remainder of 9 when divided by 6: 9 ≡ 3 (mod 6).
- Next, examine the powers of 9 modulo 6:
  - 9² = 81 ≡ 3 (mod 6)
  - 9³ = 729 ≡ 3 (mod 6)
- In general, for any integer exponent p ≥ 1, 9ᵖ ≡ 3ᵖ (mod 6). Since 3ᵖ for p ≥ 1 always results in a number whose remainder is 3 when divided by 6 (e.g., 3, 9, 27, ...), it follows that 9ᵖ ≡ 3 (mod 6).
- Therefore, each term in the series (9³, 9⁴, ..., 9¹⁰⁰) leaves a remainder of 3 when divided by 6.
Number of Terms in the Series:
- The series runs from 9³ to 9¹⁰⁰. The number of terms is calculated as (Last Exponent - First Exponent + 1) = 100 - 3 + 1 = 98 terms.
Sum of Remainders:
- Since there are 98 terms, and each term leaves a remainder of 3, the sum of these individual remainders is 98 × 3 = 294.
Final Remainder:
- To find the remainder of the entire sum when divided by 6, we find the remainder of the sum of remainders: 294 ÷ 6.
- 294 = 49 × 6 + 0.
- Thus, the final remainder is 0.

Incorrect Options:

Options 2 (1), 3 (2), and 4 (3) are incorrect. The detailed calculation using modular arithmetic demonstrates that each term in the series 9³ + 9⁴ + ... + 9¹⁰⁰ yields a remainder of 3 when divided by 6. With 98 such terms, the cumulative sum of these remainders is 294. When 294 is subsequently divided by 6, the result is 49 with a remainder of 0. This indicates that the original sum is perfectly divisible by 6, making any non-zero remainder incorrect.