10.111 Which digit should be placed in the question mark (?) position to make the resulting integer divisible by 7:
$$
\underbrace{666 \cdots 66}_{\text{so } \uparrow} \underbrace{555 \cdots 55}_{50 \uparrow} .
$$
(Leningrad Mathematical Olympiad, 1989)
［Solution］Notice that 7|111111.
Then
$$
\begin{aligned}
& \underbrace{666 \cdots 66}_{50 \uparrow} ? \underbrace{555 \cdots 55}_{\text {s0 } 1} \\
= & \underbrace{666 \cdots 66}_{48 \uparrow} \cdot 10^{53}+\overline{66 ? 55} \cdot 10^{48}+\underbrace{555 \cdots 55}_{48 \uparrow} .
\end{aligned}
$$

Since
$$
7+\underbrace{666 \cdots 66}_{48 \uparrow} \cdot 10^{53},
$$

we only need to find (?), such that
$$
71 \overline{66 ? 55} \cdot 10^{48} .
$$

Given that $\left(7,10^{48}\right)=1$, we only need to find (?) satisfying
$$
7 \text { divides } \overline{66 ? 55} \text {. }
$$

Let the digit at the (?) position be $x, x \in\{0,1,2, \cdots, 9\}$, then we have
$$
\begin{aligned}
\overline{66 \times 55} & =63035+\overline{3 \times 20} . \\
& =7 \cdot 9005+\overline{3 \times 2} \cdot 10 .
\end{aligned}
$$

It is not difficult to verify that only 322 and 392 are divisible by 7. That is
$$
7|66255,7| 66955 \text {. }
$$

Therefore, the digit (?) is 2 or 9.