यदि n धनपूर्ण संख्या है, तो (1ⁿ + 2ⁿ) को 4 से भाग देने पर उसके भिन्न शेषफलों की संख्या क्या है ?
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If n is a natural number, then what is the number of distinct remainders of (1ⁿ + 2ⁿ) when divided by 4?
(a) 0 · (b) 1 · (c) 2 · (d) 3
उत्तर और व्याख्या · explanation in English
Answer: (c) 2
1ⁿ is always 1, and 2ⁿ is a multiple of 4 once n ≥ 2. So the remainder is 3 when n = 1 and 1 for every larger n — two distinct remainders.
- n = 1: 1 + 2 = 3, remainder 3.
- n ≥ 2: 2ⁿ is divisible by 4 and 1ⁿ = 1, so 1ⁿ + 2ⁿ leaves remainder 1.
- Only two remainders occur: 3 and 1.
- Check: n = 2 gives 5 (remainder 1), n = 3 gives 9 (remainder 1).
Remember · Powers of 2 are multiples of 4 from 2² onwards; always check the first case separately.
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). ·