CAT 2023 Question Paper | Quant Slot 2

CAT Questions

CAT 2023 Quant was dominated by Algebra followed by Arithmetic. 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 1 : Let \(a, b, m\) and \(n\) be natural numbers such that \(a>1\) and \(b>1\). If \(a^m b^n=144^{145}\), then the largest possible value of \(n-m\) is

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Video Explanation

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Explanatory Answer

\( a ^ { m } b ^ { n } = 144 ^ { 145 } \)
\( a ^ { m } b ^ { n } = \left( 2 ^ { 4 } \times 3 ^ { 2 } \right) ^ { 145 } \)
\( a ^ { m } b ^ { n } = 3 ^ { 290 } \times 2 ^ { 580 } \)
Since the highest power of a prime in the prime factorization of \( 144 ^ { 145 } \) is 580, n can never be more than 580. Since m is a natural number the smallest value that m can take is 1. So, the maximum value of (n – m) is 580 – 1 = 579.
But is that possible??
Yes!
When \( a ^ { m } b ^ { n } = \left( 3 ^ { 290 } \right) ^ { 1 } \times 2 ^ { 580 } \), where \( a = 3 ^ { 290 } ; m = 1 ; b = 2 ; n = 580 \).

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The question is " Let \(a, b, m\) and \(n\) be natural numbers such that \(a>1\) and \(b>1\). If \(a^m b^n=144^{145}\), then the largest possible value of \(n-m\) is "

Hence, the answer is '579'

Choice C is the correct answer.