Civil Services Prep

Prelims 2022 · CSAT / Quantitative Aptitude · Question 74

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

  1. 7
  2. 6
  3. 5
  4. 4

Answer

6

Use the exponent of 3 in 15!:
(m = \lfloor 15/3 \rfloor + \lfloor 15/9 \rfloor = 5 + 1 = 6).
So the maximum power of 3 dividing 15 × 14 × ... × 1 = 15! is (3^6).

  • (a) 7: Would require one more factor of 3 than available in 15!. Incorrect.
  • (b) 6: Total factors of 3 in 15! are 6. Correct.
  • (c) 5: Less than the actual exponent. Incorrect.
  • (d) 4: Less than the actual exponent. Incorrect.
  1. X receives three coins of different denominations : 1, 2, 5, 10 and 20. If the total amount received by X is…
  2. For two distinct real numbers x and y, which of them is bigger? Statement I : x^2 Statement II : y
  3. If x and y are integers, then is x even? Statement I : x^2 y^2 is even. Statement II : 1 + x^2 + y^2 is odd.
  4. X is a collection of certain odd numbers whereas Y is a collection of certain even numbers. T consists of the…
  5. If x, y and z are integers, each greater than 1, then is x a prime number? Statement I : xy^2 = 116 Statement…
  6. Which of the following conclusions are valid? 1. Seven Sanskriti Angans, representing different regions of…