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

C. Given $m^n = 2^{25} 	imes 3^{40}$. Since $m$ and $n$ are natural numbers and $n > 1$, $n$ must divide both exponents 25 and 40. $$\gcd(25, 40) = 5$$ Therefore, $n = 5$. Then: $$m^5 = 2^{25} 	imes 3^{40} = (2^5 	imes 3^8)^5$$ $$m = 2^5 	imes 3^8 = 32 	imes 6561 = 209952$$ Thus, $m - n = 209952 - 5 = 209947$.

Exponent Representation and Natural Numbers

Explanation

Practise this under exam conditions