1. Find all natural numbers $n \geq 10$ with non-zero digits, for which the following holds: if we remove any digit of the number $n$, we get a number that divides $n$.
I/1. Let the number $n$ have a decimal representation $\overline{a_{k} a_{k-1} \ldots a_{2} a_{1}}$. By assumption, the number $\overline{a_{k} a_{k-1} \ldots a_{2}}$ divides the number $\overline{a_{k} a_{k-1} \ldots a_{2} a_{1}}=10 \cdot \overline{a_{k} a_{k-1} \ldots a_{2}}+a_{1}$, and thus it also divides the number $a_{1}$. Since $a_{1} \neq 0$, the number $\overline{a_{k} a_{k-1} \ldots a_{2}}$ can be at most a single-digit number, so $k=2$. If $n=\overline{a_{2} a_{1}}$, then by the above, $a_{2}$ must divide $a_{1}$, so $a_{2}=a_{1}$ or $a_{2} \leq 4$. Therefore, the number $n$ can only be $99,88,77,66,55,48,44,39,36,33,28,26,24,22,19,18,17,16,15,14,13,12$ or 11. Of these, only the numbers $99,88,77,66,55,48,44,36,33,24,22,15,12$ and 11 satisfy the condition of the problem.

In summary: all solutions are $11,12,15,22,24,33,36,44,48,55,66,77,88,99$.

![](https://cdn.mathpix.com/cropped/2024_06_07_550bffe3dbbdd9dbd37bg-06.jpg?height=60&width=1633&top_left_y=889&top_left_x=220)

![](https://cdn.mathpix.com/cropped/2024_06_07_550bffe3dbbdd9dbd37bg-06.jpg?height=57&width=1633&top_left_y=942&top_left_x=220)

![](https://cdn.mathpix.com/cropped/2024_06_07_550bffe3dbbdd9dbd37bg-06.jpg?height=51&width=1633&top_left_y=997&top_left_x=220)

All solutions (with or without justification) .......................................................

If the contestant does not list all solutions, one point is awarded for every 5 correct solutions. If the solutions include numbers that do NOT SATISFY the conditions of the problem, 1 POINT is DEDUCTED for EACH INCORRECT SOLUTION.