For positive integers n and k, let f(n, k) be the remainder when n is divided by k, and for n>1 let F(n)=\max _{1 \leq k \leq \frac{n}{2}} f(n, k). Find the remainder when \sum_{n=20}^{100} F(n) is divided by 1000 .
For positive integers n and k, let f(n, k) be the remainder when n is divided by k, and for n>1 let F(n)=\max _{1 \leq k \leq \frac{n}{2}} f(n, k). Find the remainder when \sum_{n=20}^{100} F(n) is divided by 1000 .