2022 CSAT, all questions

UPSC Prelims 2022 · CSAT

Question 74: Basic Numeracy

If 15×14×13×···×3×2×1 = 3^m × n where m and n are positive integers, then what is the maximum value of m ?

a7
b6
c5
d4

AnswerUPSC’s answer is (b).

The highest power of 3 in 15! is found by adding the multiples of 3, 9, 27 and so on: ⌊15/3⌋ + ⌊15/9⌋ = 5 + 1 = 6 (27 exceeds 15). So m is at most 6. Check: the multiples of 3 up to 15 are 3, 6, 9, 12 and 15, giving 5 threes, and the number 9 adds a sixth.

Why each option is right or wrong
  • (a) 7 would need a further factor of 3 that 15! does not have.
  • (b) Correct: 5 + 1 = 6.
  • (c) 5 counts only the multiples of 3, missing the extra 3 in 9.
  • (d) 4 undercounts the factors of 3.

Answer: UPSC’s key. Explanation: written by UPSC Grade, not by UPSC.

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Source: UPSC, Civil Services (Preliminary) Examination 2022, General Studies Paper II (CSAT), Series A. The paper on upsc.gov.in.