2010 AMC 10B Problem 18

Positive integers a, b, and c are randomly and independently selected with replacement from the set \{1, 2, 3,\ldots, 2010\}. What is the probability that abc + ab + a is divisible by 3?

\textbf{(A)}\ \dfrac{1}{3} \qquad \textbf{(B)}\ \dfrac{29}{81} \qquad \textbf{(C)}\ \dfrac{31}{81} \qquad \textbf{(D)}\ \dfrac{11}{27} \qquad \textbf{(E)}\ \dfrac{13}{27}

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