For each positive integer p, let b(p) denote the unique positive integer k such that |k-\sqrt{p}|<\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 .
For each positive integer p, let b(p) denote the unique positive integer k such that |k-\sqrt{p}|<\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 .