Question 20
Let $m$, $n$ and $a_1, a_2, \dots, a_m$ be arbitrary positive integers. Ali and Mohammad play the following game: At each step, Ali chooses $b_1, b_2, \dots, b_m \in \mathbb{N}$, then Mohammad chooses a positive integer $s$ and obtains a new sequence $\{c_i = a_i + b_{i+s}\}_{i=1}^m$, where
$$
b_{m+1} = b_1,\ b_{m+2} = b_2,\ \dots,\ b_{m+s} = b_s.
$$
Through a finite number of steps, Ali intends to make all the numbers divisible by $n$. Find all positive integers $m$ and $n$ such that Ali, independent of the initial values $a_1, a_2, \dots, a_m$, has a winning strategy.
$$
b_{m+1} = b_1,\ b_{m+2} = b_2,\ \dots,\ b_{m+s} = b_s.
$$
Through a finite number of steps, Ali intends to make all the numbers divisible by $n$. Find all positive integers $m$ and $n$ such that Ali, independent of the initial values $a_1, a_2, \dots, a_m$, has a winning strategy.