CAT 2024 Slot 2 QA Q3 Logarithms & Exponents

Question 3 of the Quantitative Ability (QA) section, CAT 2024 Slot 2, with CAT's official answer and a worked solution.

If \(m\) and \(n\) are natural numbers such that \(n>1\), and \(m^n=2^{25} \times 3^{40}\), then \(m-n\) equals
  1. A
    209932
  2. B
    209937
  3. C
    209947
  4. D
    209942

Answer: (C) 209947 Logarithms & Exponents MCQ

Solution

  1. For \(m^n = 2^{25} \cdot 3^{40}\) with natural \(m\), \(n\) must divide both 25 and 40, so \(n \mid 5\). Since \(n>1\), \(n=5\).
  2. \(m = 2^5 \cdot 3^8 = 32 \times 6561 = 209952\).
  3. \(m-n = 209952 - 5 = 209947\).

Shortcut. Only the units digit matters: \(m\) ends in 2 (\(32 \times 6561\)), so \(m-5\) ends in 7; then \(m\) itself pins the option.

Same concept in other years

See every question on Writing a number as m^n: n must divide every prime exponent.

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