# Problem 5. (3 points)

From a natural number, its largest digit was subtracted. Then, from the resulting number, its largest digit was also subtracted, and so on. After five such operations, the number 200 was obtained. What number could have been written initially? List all the options and prove that there are no others.
Answer: All numbers from 222 to 229

Solution:

For numbers from 201 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.

| 201 | 199 | - |
| :---: | :---: | :---: |
| $20 x, x>1$ | 200 | 1 |
| 210 | 108 | 2 |
| 211 | 109 | 2 |
| $21 x, x>1$ | 210 | 3 |
| 220 | 218 | 4 |
| 221 | 219 | 4 |
| $22 x, x>1$ | 220 | 5 |

From all subsequent numbers, the result of the first operation is no less than 220, which means it takes more than 5 operations to get to 100, if it is possible at all.