Problem 8.4. Find any solution to the puzzle

$$
\overline{A B C A}=182 \cdot \overline{C D}
$$

where $A, B, C, D$ are four different non-zero digits (the notation $\overline{X Y \ldots Z}$ means the decimal representation of the number).

As an answer, write the four-digit number $\overline{A B C D}$.
Answer: 2916.

Solution. $\overline{A B C A}=1001 A+10 \cdot \overline{B C}$. Note that 1001 and 182 are divisible by 91, therefore, $10 \cdot \overline{B C}$ is also divisible by 91, i.e., $\overline{B C}=91$.

Substituting $B=9, C=1$ and dividing by 91, we get the equation $11 A=10+2 D$. On the left side of this equation, we have a number divisible by 11, and on the right side, an even two-digit number less than 30. Therefore, this number is 22, from which $A=2$ and $D=6$.