jmc

$$\begin{array}{rl}\frac{1}{p}+\frac{1}{q}+\frac{1}{r}+\frac{360}{pqr}& =1\\ pq+pr+qr+360& =pqr\\ 360& =pqr-pq-pr-qr\\ & =(p-1)(q-1)(r-1)-(p+q+r)+1\\ & =(p-1)(q-1)(r-1)-25\\ 385& =(p-1)(q-1)(r-1)\end{array}$$ From here, you can factor $385$ as $5\cdot 7\cdot 11$, giving corresponding values of $6,8,$ and $12$. The answer is $6\cdot 8\cdot 12=\boxed{576}$.