2. Petya wrote a natural number on the board, and Vasya erased the first two digits of it. As a result, the number decreased by 149 times. What could Petya's number be, given that it is odd?
Answer: 745 or 3725.

Solution. Petya's number can be represented as $10^{k} a+b$, where $10 \leqslant a \leqslant 99, b<10^{k}$. According to the condition, $10^{k} a+b=149 b$. From this, $10^{k} a=37 \cdot 4 b$. Since $10^{k}$ and 37 are coprime numbers, the number $a$ must be divisible by 37. Among two-digit numbers, only 37 and 74 have this property.

If $a=37$, then $10^{k}=4 b$. Since $b$ is odd, $10^{k}$ is divisible by 4 but not by 8. Therefore, $k=2, b=25$, and the desired number is 3725.

If $a=74$, then $10^{k}=2 b$. Since $b$ is odd, $10^{k}$ is divisible by 2 but not by 4. Therefore, $k=1, b=5$, and the desired number is 745.

Grading. 13 points for a correct solution. For each found answer without justification that there are no other solutions, 2 points.