Selection and Training Session

Let $p\ge 2$ be a prime number. Alice and Bob play the following game: they, in turn, select an index $i$ in the set $\{0,1,2,\dots ,p-1\}$ that was not selected before by either of the two players and then chooses a digit $a_i$. Alice starts. The game ends after all the indices have been selected. The goal of Alice is to make the number $$M=a_0+10\cdot a_1+10^2\cdot a_2+\cdots +10^{p-1}{a}_{p-1}$$ divisible by $p$, and the goal of Bob is to prevent this. Prove that Alice has the winning strategy.