AMC 12 2010a Problem 20

Arithmetic sequences $\left(a_n\right)$ and $\left(b_n\right)$ have integer terms with $a_1=b_1=1<a_2 \le b_2$ and $a_n b_n = 2010$ for some $n$. What is the largest possible value of $n$?

\textbf{(A)}\ 2 \textbf{(B)}\ 3 \textbf{(C)}\ 8 \textbf{(D)}\ 288 \textbf{(E)}\ 2009