UPSC Prelims 2026 · CSAT
Question 19: Basic Numeracy
If 10^m × 1000 × n = 75^25 × 25^32 × 32^75, where n is not divisible by 10, then the value of m is
AnswerUPSC’s answer is (b).
Factorise the right side: 75^25 = 3^25 × 5^50, 25^32 = 5^64 and 32^75 = 2^375, so the product is 2^375 × 3^25 × 5^114. Each 10 needs one 2 and one 5, and there are only 114 fives, so at most 114 tens can be formed. Since n has no factor of 10, the factor 10^m × 1000 = 10^(m+3) must carry all of them: m + 3 = 114, so m = 111.
Why each option is right or wrong
- (a) Wrong: m = 101 gives only 104 tens, leaving a factor of 10 inside n.
- (b) Right: m + 3 = 114 gives m = 111.
- (c) Wrong: m = 121 needs 124 tens, but only 114 fives exist.
- (d) Wrong: m = 131 needs 134 tens, but only 114 fives exist.
Only the provisional key (27 May 2026) was found; the final key replaces it when found.
Answer: UPSC’s key. Explanation: written by UPSC Grade, not by UPSC.
Source: UPSC, Civil Services (Preliminary) Examination 2026, General Studies Paper II (CSAT), Series A. The paper on upsc.gov.in.