jmc

The prime factorization of $27000$ is $2^3\cdot 3^3\cdot 5^3.$ These three factors, $2^3,$ $3^3,$ and $5^3$ are pairwise relatively prime, and this is the only possible triple of positive integers satisfying the given conditions. Therefore, the answer is $$2^3+3^3+5^3=8+27+125=\boxed{160}.$$