CAT 2023 Question Paper | Quant Slot 2

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

Video Explanation

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 $a = 3 ^ { 290 } ; m = 1 ; b = 2 ; n = 580$.

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.

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