9-5-1. The numbers 1 and 7 are written on the board. In one move, the greatest common divisor of both numbers on the board is added to each of them. For example, if at some point the numbers 20 and 50 are on the board, they will be replaced by the numbers 30 and 60.

What numbers will be on the board after 100 moves? Provide the answers in any order. Answer. 588 and 594.
Solution option 1. Let's perform the first few actions described in the condition.

| actions | first number | second number | their GCD |
| :---: | :---: | :---: | :---: |
| none | 1 | 7 | 1 |
| after the first | 2 | 8 | 2 |
| after the second | 4 | 10 | 2 |
| after the third | 6 | 12 | 6 |
| after the fourth | 12 | 18 | 6 |

Notice that with all subsequent actions, 6 will be added to both numbers, since the difference between the numbers will always be 6, and each of them will be divisible by 6.

Since after the third action the first number is 6, and we will perform 97 more actions with it, it will become equal to $98 \cdot 6 = 588$, and the second number will then be equal to $588 + 6 = 594$.

Comment. It can be proven that, starting with any pair of numbers $a \neq b$, we will eventually get both numbers divisible by $a - b$.