2013 AMC 12A Problem 10

Let S be the set of positive integers n for which \tfrac{1}{n} has the repeating decimal representation 0.\overline{ab} = 0.ababab\cdots, with a and b different digits. What is the sum of the elements of S?

\textbf{(A)}\ 11\qquad\textbf{(B)}\ 44\qquad\textbf{(C)}\ 110\qquad\textbf{(D)}\ 143\qquad\textbf{(E)}\ 155\qquad

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