Question 36
Let $p$ be a prime. In the following game, Ed and Afei take turns. On a player's turn, he first chooses as $i$ one of the numbers in $\{0, 1, \dots, p-1\}$ that has not yet been chosen by either player, and then chooses an element of $\{0, 1, 2, \dots, 9\}$ as $a_i$. Ed moves first. When all of $\{0, 1, \dots, p-1\}$ have been chosen, the game ends and the following number is computed:
$$
M = a_0 + 10 \times a_1 + \cdots + 10^{p-1} \times a_{p-1} = \sum_{j=0}^{p-1} a_j \times 10^j.
$$
If $M$ is divisible by $p$, Ed wins; otherwise, Afei wins.
Prove that Ed has a winning strategy.
$$
M = a_0 + 10 \times a_1 + \cdots + 10^{p-1} \times a_{p-1} = \sum_{j=0}^{p-1} a_j \times 10^j.
$$
If $M$ is divisible by $p$, Ed wins; otherwise, Afei wins.
Prove that Ed has a winning strategy.