# Problem 5. (3 points)

A natural number had its largest digit subtracted from it. Then, from the resulting number, its largest digit was also subtracted, and so on. After six such operations, the number 100 was obtained. What number could have been written initially? List all the options and prove that there are no others.
Answer: $130,131,132$

## Solution:

For numbers from 101 to 139, we will show which number they transition to. In the third column, it is indicated how many operations it takes to get to 100.

| $10 x, x>0$ | 100 | 1 |
| :---: | :---: | :---: |
| 110 | 109 | 2 |
| $11 x, x>0$ | 110 | 3 |
| 120 | 118 | 4 |
| 121 | 119 | 4 |
| $12 x, x>1$ | 120 | 5 |
| 130 | 127 | 6 |
| 131 | 128 | 6 |
| 132 | 129 | 6 |

From all other numbers, after the first operation, the result is no less than 130, meaning it takes more than 6 operations to get to 100, if it is possible at all.