9³ + 9⁴ + 9⁵ + 9⁶ + . . . + 9¹⁰⁰ को 6 से भाग देने पर शेषफल क्या है ?
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What is the remainder when 9³ + 9⁴ + 9⁵ + 9⁶ + ... + 9¹⁰⁰ is divided by 6?
(a) 0 · (b) 1 · (c) 2 · (d) 3
उत्तर और व्याख्या · explanation in English
Answer: (a) 0
Every power of 9 leaves remainder 3 when divided by 6, and there are 98 terms. 98 × 3 = 294 is a multiple of 6, so the remainder is 0.
- 9 leaves remainder 3 when divided by 6, and 3 × 3 = 9 again leaves 3. So every power 9ᵏ leaves remainder 3.
- Number of terms from 9³ to 9¹⁰⁰: 100 − 3 + 1 = 98.
- Sum of remainders = 98 × 3 = 294 = 6 × 49, so the remainder is 0.
- Check: each term is odd and a multiple of 3; 98 odd terms add to an even number, and an even multiple of 3 is divisible by 6.
Remember · For the remainder of a sum, reduce each term first; count terms carefully (3 to 100 is 98 terms).
Question and answer: UPSC's official GS Paper II (2025, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). ·