MathDB

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}

and then Mohammad chosses a positive integers

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

The goal of Ali is to make all the numbers divisible by

n

in a finite number of steps. FInd all positive integers

m

and

n

such that Ali has a winning strategy, no matter how the initial values

a_1, a_2,\dots,a_m

are. after we create the

c_i

s, this sequence becomes the sequence that we continue playing on, as in it is our 'new'

a_i

Proposed by Shayan Gholami

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