If 10ᵐ × 1000 × n = 75²⁵ × 25³² × 32⁷⁵, where n is not divisible by 10, then the value of m is
- (a) 101
- (b) 111 ✓ UPSC's answer
- (c) 121
- (d) 131
Why the answer is (b)
• Write each base in primes: 75 = 3 × 5², 25 = 5², 32 = 2⁵.
• So 75²⁵ × 25³² × 32⁷⁵ = 3²⁵ × 5⁵⁰ × 5⁶⁴ × 2³⁷⁵ = 3²⁵ × 5¹¹⁴ × 2³⁷⁵.
• The number of factors of 10 is the smaller of the powers of 2 and 5, i.e. 114. What is left, n = 3²⁵ × 2²⁶¹, is not divisible by 10.
• 10ᵐ × 1000 = 10ᵐ⁺³, so m + 3 = 114 and m = 111: (b).
Why the other options are wrong
- (a) 101
- 101 undercounts the powers of 5 (there are 114).
- (c) 121
- 121 is too large; the power of 10 available is 114, and 3 of it is the 1000.
- (d) 131
- 131 is too large; only 114 factors of 10 are available.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2026, held on 24 May 2026. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.