3. Determine the three-digit number $\overline{a b c}$ if $\overline{a b} \cdot \overline{c c} \cdot \overline{a b c}=\overline{a b c a b c}$.
3. It holds that $\overline{a b} \cdot \overline{c c} \cdot \overline{a b c}=\overline{a b c a b c}$

$$
\begin{aligned}
& \overline{a b} \cdot \overline{c c} \cdot \overline{a b c}=\overline{a b c 000}+\overline{a b c} \\
& \overline{a b} \cdot \overline{c c} \cdot \overline{a b c}=1000 \cdot \overline{a b c}+\overline{a b c} \\
& \overline{a b} \cdot \overline{c c} \cdot \overline{a b c}=1001 \cdot \overline{a b c}
\end{aligned}
$$

By dividing the equation by $\overline{a b c}$, we get $\overline{a b} \cdot \overline{c c}=1001$

$$
\overline{a b} \cdot \overline{c c}=7 \cdot 11 \cdot 13
$$

Finally, $\overline{a b} \cdot \overline{c c}=13 \cdot 77$ or $\overline{a b} \cdot \overline{c c}=91 \cdot 11$.

Therefore, $\overline{a b c}=137$ or $\overline{a b c}=911$.