\begin{eqnarray*}S & = & \sum_{k = 0}^{m - 1}\sum_{r = 0}^{n - 1}{(mn + 1)k + mr\over mn} \\
& = & \sum_{k = 0}^{m - 1}\left(n\cdot{(mn + 1)k\over mn} + {1\over n}\cdot{n(n - 1)\over 2}\right) \\
& = & \sum_{k = 0}^{m - 1}\left({(mn + 1)k\over m} + {n - 1\over 2}\right) \\
& = & {mn + 1\over m}\cdot {m(m - 1)\over 2} + {m(n - 1)\over 2} \\
& = & {(mn + 1)(m - 1) + m(n - 1)\over 2} \\
& = & {m^2n - mn + m - 1 + mn - m\over 2} \\
& = & {m^2n - 1\over 2}\end{eqnarray*}