2010 AMC 12A 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 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 8 \qquad \textbf{(D)}\ 288 \qquad \textbf{(E)}\ 2009

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.