2022 AMC 10A Problem 17

How many three-digit positive integers \underline{a} \ \underline{b} \ \underline{c} are there whose nonzero digits a,b, and c satisfy 0.\overline{\underline{a}~\underline{b}~\underline{c}} = \frac{1}{3} (0.\overline{a} + 0.\overline{b} + 0.\overline{c})? (The bar indicates repetition, thus 0.\overline{\underline{a}~\underline{b}~\underline{c}} is the infinite repeating decimal 0.\underline{a}~\underline{b}~\underline{c}~\underline{a}~\underline{b}~\underline{c}~\cdots)

\textbf{(A) } 9 \qquad \textbf{(B) } 10 \qquad \textbf{(C) } 11 \qquad \textbf{(D) } 13 \qquad \textbf{(E) } 14

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