यदि 10ᵐ × 1000 × n = 75²⁵ × 25³² × 32⁷⁵ है, जहाँ n, 10 से विभाज्य नहीं है, तो m का मान क्या है?
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If 10ᵐ × 1000 × n = 75²⁵ × 25³² × 32⁷⁵, where n is not divisible by 10, then the value of m is
(a) 101 · (b) 111 · (c) 121 · (d) 131
उत्तर और व्याख्या · explanation in English
Answer: (b) 111
Break the right side into primes: 2³⁷⁵ × 3²⁵ × 5¹¹⁴. The number of 10s is the smaller of the powers of 2 and 5, i.e., 114, and n must hold none of them, so m + 3 = 114.
- 75²⁵ = (3 × 5²)²⁵ = 3²⁵ × 5⁵⁰.
- 25³² = 5⁶⁴.
- 32⁷⁵ = (2⁵)⁷⁵ = 2³⁷⁵.
- Right side = 2³⁷⁵ × 3²⁵ × 5¹¹⁴, which contains exactly 10¹¹⁴ (limited by the 5s).
- n = 2²⁶¹ × 3²⁵ is not divisible by 10, so 10ᵐ × 10³ = 10¹¹⁴ and m = 111.
Remember · Powers of 10 in a product = min(power of 2, power of 5). Count both, take the smaller.
Question and answer: UPSC's provisional GS Paper II (2026, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). ·