Question 11
For each positive integer $p,$ let $b(p)$ denote the unique positive integer $k$ such that $|k - \sqrt{p}| \lt \frac{1}{2}.$ For example, $b(6) = 2$ and $b(23) = 5.$ If $S = \sum_{p=1}^{2007} b(p),$ find the remainder when $S$ is divided by $1000.$