CAT 2023 Question Paper | Quant Slot 3

CAT 2023 Quant was dominated by Arithmetic followed by Algebra. In Arithmetic, the questions were dominated by topics like Speed-time-distance, Mixture and Alligations. This year, there was a surprise. The questions from Geometry were relatively on the lower side as compared to the previous years. There were 8 TITA Qs this year. Overall this section was at a medium level of difficulty.

Question 2 : Let \(n\) and \(m\) be two positive integers such that there are exactly 41 integers greater than \(8^m\) and less than \(8^n\), which can be expressed as powers of 2 . Then, the smallest possible value of \(n+m\) is

Video Explanation

Explanatory Answer

We need to have 41 integers that can be expressed as powers of 2 between 8^m and 8ⁿ.
That is we need to have 41 integers that can be expressed as powers of 2 between 2^3m and 2³ⁿ.
The numbers will be of the form: \( 2 ^ { 3 m } , 2 ^ { 3 n + 1 } , 2 ^ { 3 m + 2 } , 2 ^ { 3 m + 3 } , \ldots , 2 ^ { 3 m + 41 } , 2 ^ { 3 n } \)
clearly, \( \quad 3 n - 1 = 3 m + 41 \)
\( 3 ( n - m ) = 42 \)
\( n - m = 14 \)
The smallest value \( m \) can take \( = 1 \), then \( n = 15 \)
\( m + n = 1 + 15 = 16 \)

The question is " Let \(n\) and \(m\) be two positive integers such that there are exactly 41 integers greater than \(8^m\) and less than \(8^n\), which can be expressed as powers of 2 . Then, the smallest possible value of \(n+m\) is "

The answer is '16'

Choice C is the correct answer.

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