Q. What is the remainder when 9³ + 9⁴ + 9⁵ + 9⁶ + … + 9¹⁰⁰ is divided by 6?
(a) 0
(b) 1
(c) 2
(d) 3
UPSC Prelims 2025 CSAT
Explanation :
1. Note that 9 ≡ 3 (mod 6).
2. Then for any k ≥ 1,
9^k ≡ 3^k (mod 6).
3. But 3^1 ≡ 3 (mod 6), and multiplying by 3 again gives
3^2 = 9 ≡ 3 (mod 6),
3^3 = 27 ≡ 3 (mod 6),
and so on.
By induction, 3^k ≡ 3 (mod 6) for every k ≥ 1.
4. The sum 9³ + 9⁴ + … + 9¹⁰⁰ has 100 – 3 + 1 = 98 terms, each ≡ 3 (mod 6).
5. Total ≡ 98 × 3 = 294 ≡ 0 (mod 6).
