AMC 12 2013a 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 \textbf{(B)}\ 44 \textbf{(C)}\ 110 \textbf{(D)}\ 143 \textbf{(E)}\ 155