imc

$3\cdot 5^{2}m$ must be a perfect cube, so each power of a prime in the factorization for $3\cdot 5^{2}m$ must be divisible by $3$. Thus the minimum value of $m$ is $3^2\cdot 5=45$, which makes $n=\sqrt[3]{3^3\cdot 5^3}=15$. The minimum possible value for the sum of $m$ and $n$ is $\boxed{60}.$