Which positive integer \(n\) satisfies the equation \[\frac{3}{n^3}+\frac{4}{n^3}+\frac{5}{n^3}+\dotsb +\frac{n^3-5}{n^3}+\frac{n^3-4}{n^3}+\frac{n^3-3}{n^3}=60?\]

Which positive integer \(n\) satisfies the equation \[\frac{3}{n^3}+\frac{4}{n^3}+\frac{5}{n^3}+\dotsb +\frac{n^3-5}{n^3}+\frac{n^3-4}{n^3}+\frac{n^3-3}{n^3}=60?\]