Question 14
For positive integers $n$ and $k,$ let $f(n, k)$ be the remainder when $n$ is divided by $k,$ and for $n \gt 1$ let $$
F(n) = \max_{1 \le k \le \frac{n}{2}} f(n, k).
$$ Find the remainder when $\sum_{n = 20}^{100} F(n)$ is divided by $1000.$
F(n) = \max_{1 \le k \le \frac{n}{2}} f(n, k).
$$ Find the remainder when $\sum_{n = 20}^{100} F(n)$ is divided by $1000.$