MathNet 2008-part-05 — 164 problems (EN–VI) Question 29

For a positive integer $n$, let $\theta(n)$ denote the number of integers $0 \leq x \lt 2010$ such that $x^{2} - n$ is divisible by $2010$. Determine the remainder when $\sum_{n=0}^{2009} n \cdot \theta(n)$ is divided by $2010$.