CAT 2023 Slot 1 QA Q4 Number Systems

Question 4 of the Quantitative Ability (QA) section, CAT 2023 Slot 1, with CAT's official answer and a worked solution.

Let \(n\) be the least positive integer such that 168 is a factor of \(1134^n\). If \(m\) is the least positive integer such that \(1134^n\) is a factor of \(168^m\), then \(m+n\) equals
  1. A
    12
  2. B
    9
  3. C
    15
  4. D
    24

Answer: (C) 15 Number Systems MCQ

Solution

  1. \(1134 = 2\cdot 3^4\cdot 7\) and \(168 = 2^3\cdot 3\cdot 7\).
  2. For \(168 \mid 1134^n\) we need \(2^n \ge 2^3\), so \(n = 3\).
  3. \(1134^3 = 2^3\cdot 3^{12}\cdot 7^3\) must divide \(168^m = 2^{3m}3^m7^m\); the power of 3 forces \(m \ge 12\), so \(m=12\).
  4. \(m+n = 15\).

Same concept in other years

See every question on Equate exponents of each prime on both sides.

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