cat 2024 Question 15

Question

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

Accepted Answer

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 $\left(2^5 ight)^5 imes\ \left(3^8 ight)^5$

$\left(32 imes\ 81 imes\ 81 ight)^5$

$\left(209952 ight)^5$

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

Therefore, Option D is the correct answer.

Question 15.