If 10ᵐ × 1000 × n = 75²⁵ × 25³² × 32⁷⁵, where n is not divisible by 10, then the value of m is

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.

1. 75²⁵ = (3 × 5²)²⁵ = 3²⁵ × 5⁵⁰.
2. 25³² = 5⁶⁴.
3. 32⁷⁵ = (2⁵)⁷⁵ = 2³⁷⁵.
4. Right side = 2³⁷⁵ × 3²⁵ × 5¹¹⁴, which contains exactly 10¹¹⁴ (limited by the 5s).
5. 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). · Report an error