2. Alyona erased one digit from a five-digit number divisible by 99, and it turned out that the resulting four-digit number is also divisible by 99. Which digit in order could have been erased? (Neither the original nor the resulting number can start with zero.)
Solution. Answer: any, except the first and second Examples:

$$
\begin{aligned}
99099 & \rightarrow 9999 \\
49500 & \rightarrow 4950 \\
99990 & \rightarrow 9999
\end{aligned}
$$

First, note that according to the divisibility rule for 9, the digit 0 or 9 was removed.

The first digit cannot be removed: since it is not 0, the absolute difference between the sums of the digits in even and odd positions will change, but not by more than this digit itself, which will alter the remainder when the number is divided by 11.

Now suppose from $\overline{a b c d e}$ (where $a+c+e-b-d: 11$) we removed the second digit, resulting in $\overline{a c d e}$, in which $a+d-c-e: 11$. Adding these expressions, we get $2 a-b: 11$, which is impossible for $b=0$ or $b=9$.