ऐसे कितने धन पूर्णांक हैं जिनसे 1186 को विभाजित करने पर शेषफल 31 आता है ?
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How many natural numbers are there which give a remainder of 31 when 1186 is divided by these natural numbers?
(a) 6 · (b) 7 · (c) 8 · (d) 9
उत्तर और व्याख्या · explanation in English
Answer: (d) 9
If 1186 leaves remainder 31, the divisor divides 1186 − 31 = 1155 and must be greater than 31. 1155 = 3 × 5 × 7 × 11 has 16 divisors, of which 9 are greater than 31.
- 1186 = n × (quotient) + 31, so n divides 1186 − 31 = 1155; also n > 31, because a remainder is always less than the divisor.
- 1155 = 3 × 5 × 7 × 11, so it has 2 × 2 × 2 × 2 = 16 divisors.
- Divisors greater than 31: 33, 35, 55, 77, 105, 165, 231, 385, 1155 — 9 numbers.
Remember · Remainder r from N ÷ n means n divides N − r and n > r; count divisors of N − r above r.
Question and answer: UPSC's official GS Paper II (2023, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). ·