5. Aneta and Viktor are playing a game. The game starts with Aneta saying a number from 1 to 7. Then Viktor adds that number to another number from 1 to 7 and says the sum. Aneta, in turn, adds the number Viktor said to another number from 1 to 7 and says the sum, and this process alternates until one of them says the number 100. The winner is the one who first says the number 100. Can Aneta, as the player who starts the game, ensure a sure victory, and how?
Solution. Aneta will win this game if she reaches the number 92 (then Victor would get 93, 94, ..., 98, or 99, and clearly Aneta, adding 7, 6, ..., 2, or 1, would say the number 100). To definitely say the number 92, in the previous step, Aneta needs to say the number 84, or previously say 76, etc. Since $92=11 \cdot 8+4$, Aneta, as the first player, needs to start with the number 4 and in each of her subsequent turns say the numbers less than one hundred that when divided by 8 give a remainder of 4, thus securing a sure win.

## VII Section