Question 1.

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 C