Mary and Pat play the following number game. Mary picks an initial integer greater than 2017. She then multiplies this number by 2017 and adds 2 to the result. Pat will add 2019 to this new number and it will again be Mary’s turn. Both players will continue to take alternating turns. Mary will always multiply the current number by 2017 and add 2 to the result when it is her turn. Pat will always add 2019 to the current number when it is his turn. Pat wins if any of the numbers obtained by either player is divisible by 2018. Mary wants to prevent Pat from winning the game.
Determine, with proof, the smallest initial integer Mary could choose in order to achieve this. game strategygamenumber theoryminimum