Question 58
Find the digits $A$, $B$, $C$ such that the given multiplication procedure is correct:
$$
\begin{array}{c}
\overline{ABC} \cdot \overline{BAC} \\
\multicolumn{2}{c}{---} \\
--A & \\
\multicolumn{2}{c}{---B} \\
\end{array}
$$
Where $\overline{ABC} \cdot C = \overline{---A}$, $\overline{ABC} \cdot A = \overline{---A}$, $\overline{ABC} \cdot B = \overline{---B}$. Distinct letters denote distinct digits and every "-" denotes one digit.
$$
\begin{array}{c}
\overline{ABC} \cdot \overline{BAC} \\
\multicolumn{2}{c}{---} \\
--A & \\
\multicolumn{2}{c}{---B} \\
\end{array}
$$
Where $\overline{ABC} \cdot C = \overline{---A}$, $\overline{ABC} \cdot A = \overline{---A}$, $\overline{ABC} \cdot B = \overline{---B}$. Distinct letters denote distinct digits and every "-" denotes one digit.