CAT 2023

Slot 2

QA

Question & Solution

It is given that $a^m\cdot b^n\ =\ 144^{145}$, where $a>1$ and $b>1$.

$144$ can be written as $144\ =\ 2^4\times\ 3^2$

Hence, $a^m\cdot b^n\ =\ 144^{145}$ can be written as $a^m\cdot b^n\ =\ \left(2^4\times\ 3^2\right)^{^{145}}=2^{580}\times\ 3^{290}$

We know that $3^{290}$ is a natural number, which implies it can be written as $a^1$, where $a\ >\ 1$

Hence, the least possible value of m is 1. Similarly, the largest value of n is 580.

Hence, the largest value of (n-m) is (580-1) = 579

The correct option is D