7.4. Petya and the Wolf are playing a game. Initially, the number 0 is written on the board, and on each turn, the written number must be increased by 2, 3, 8, or 9. Moves are made alternately, with Petya going first. The player who, after their move, makes the number a multiple of 2020 wins. Who can ensure their victory regardless of the opponent's actions?
Answer: Petya.

Solution: Let Petya first write 3, and then complement the Wolf's move so that the sum of their moves adds up to 11 (if the Wolf adds 3, then Petya adds 8, and vice versa; if the Wolf adds 2, then Petya adds 9, and vice versa). Thus, after Petya's move, the number will be of the form $3 + 11k$. In this way, no number of this form will be skipped, as the Wolf and Petya add exactly 11. After the Wolf's move, the number can be of one of the following forms: $11k$ (if the Wolf adds 8), $11k + 5$ (if the Wolf adds 2), $11k + 6$ (if the Wolf adds 3), $11k + 1$ (if the Wolf adds 9). Note that $2020 = 183 * 11 + 7$. No player can obtain this. Further, $4040 = 367 * 11 + 3$. A number of this form cannot be obtained by the Wolf, but Petya must obtain it, as shown earlier.

## Criteria:

Only answer - 0 points.

Highlighting paired moves 3-8, 2-9 (even if the strategy is for the other player), no other progress - 2 points.