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

Quantitative Aptitude ·Previously asked in UPSC Civil Services Examination 2026

View the full solved paper: GS Paper II

Question

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

  1. A. 101
  2. B. 111 (Correct answer)
  3. C. 121
  4. D. 131

Correct Answer

Option B — 111

Detailed Solution & Explanation

The correct answer is Option B (111).

Key Points

  • Factorise the right side: 75²⁵ = 3²⁵·5⁵⁰, 25³² = 5⁶⁴, 32⁷⁵ = 2³⁷⁵. So RHS = 2³⁷⁵ · 3²⁵ · 5¹¹⁴.
  • The number of factors of 10 = min(power of 2, power of 5) = min(375, 114) = 114. So RHS = 10¹¹⁴ × (2²⁶¹ · 3²⁵), and the bracket is not divisible by 10.
  • On the left, 10ᵐ × 1000 = 10^(m+3), so 10^(m+3) × n matches 10¹¹⁴ × (non-multiple of 10) ⇒ m + 3 = 114 ⇒ m = 111.

Additional Information

  • Pulling out all the 10s (limited by the smaller of the 2- and 5-powers) fixes m → Option B.