CAT 2024 Slot 2 QA Question 21: If $m$ and $n$ are natural numbers such that $n 1$, and $m^n = 2^{25} imes 3^{40}$, then $m - n$ equals

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

For $m^n$ to equal $2^{25} 	imes 3^{40}$, $n$ must divide both exponents $25$ and $40$.
$\gcd(25,40) = 5$
Since $n > 1$, the only possible value is $n = 5$.
$m^5 = 2^{25} 	imes 3^{40}$
$\Rightarrow m = (2^5)(3^8)$
$\Rightarrow m = 32 	imes 6561 = 209952$
$m - n = 209952 - 5 = 209947$
Answer: (2)

If mmm and nnn are natural numbers such that n>1n > 1n>1, and mn=225×340m^n = 2^{25} \times 3^{40}mn=225×340, then m−nm - nm−n equals

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