MathDB

Let

p \geq 2

be a prime number. Eduardo and Fernando play the following game making moves alternately: in each move, the current player chooses an index

i

in the set

\{0,1,2,\ldots, p-1 \}

that was not chosen before by either of the two players and then chooses an element

a_i

from the set

\{0,1,2,3,4,5,6,7,8,9\}

. Eduardo has the first move. The game ends after all the indices have been chosen .Then the following number is computed:

M=a_0+a_110+a_210^2+\cdots+a_{p-1}10^{p-1}= \sum_{i=0}^{p-1}a_i.10^i

. The goal of Eduardo is to make

M

divisible by

p

, and the goal of Fernando is to prevent this.
Prove that Eduardo has a winning strategy.
Proposed by Amine Natik, Morocco

N2