# 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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