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Repeating Decimals Timesink

Source: 2022 AMC 10A #17

November 11, 2022
AMCAMC 102022 AMC 10a2022 AMCnumber theory

Problem Statement

How many three-digit positive integers a‾\underline{a} b‾\underline{b} c‾\underline{c} are there whose nonzero digits aa, bb, and cc satisfy
0.a‾ b‾ c‾‾=13(0.a‾+0.b‾+0.c‾)?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.a‾ b‾ c‾‾0.\overline{\underline{a}~\underline{b}~\underline{c}} in the infinite repeating decimal 0.a‾ b‾ c‾ a‾ b‾ c‾ ⋯0.\underline{a}~\underline{b}~\underline{c}~\underline{a}~\underline{b}~\underline{c}~\cdots)
<spanclass=′latex−bold′>(A)</span>9<spanclass=′latex−bold′>(B)</span>10<spanclass=′latex−bold′>(C)</span>11<spanclass=′latex−bold′>(D)</span>13<spanclass=′latex−bold′>(E)</span>14<span class='latex-bold'>(A) </span>9\qquad<span class='latex-bold'>(B) </span>10\qquad<span class='latex-bold'>(C) </span>11\qquad<span class='latex-bold'>(D) </span>13\qquad<span class='latex-bold'>(E) </span>14
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