CAT 2024 Slot 2 QA Question & Solution

Number SystemsMedium

Question

If $m$ and $n$ are natural numbers such that $n > 1$, and $m^n = 2^{25} \times 3^{40}$, then $m - n$ equals

Options

209932

209937

209942

209947

Solution

We must bring the right-hand side in the form so that everything has the same power.

25 has factors 1, 5 and 25
The only common factor 40 and 25 have is 5 (other than 1 of course, which does not work)

So the right-hand side can be rewritten as $\left(2^5\right)^5\times\ \left(3^8\right)^5$
$\left(32\times\ 81\times\ 81\right)^5$
$\left(209952\right)^5$

Giving the value of m - n as 209952 - 5 = 209947

Therefore, Option D is the correct answer.