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