For Each positive integer p

For every positive integer \(p\), let \(b(p)\) denote the unique positive integer \(k\) such that \[|k-\sqrt{p}| < \dfrac{1}{2}.\]

If \(S = \displaystyle\sum_{p=1}^{2007} b(p)\) find the remainder when \(S\) is divided by 1000.

For example, \(b(6) = 2\) and \( b(23) = 5\).

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