Alice and Bob play a game starting from Alice where Alice and Bob are supposed to write a number to the board, Then the game goes through several rounds between Alice and Bob with the rules below:
- Alice goes first and writes on the board a positive integer less than or equal to our top-limit number $2017$. After this, neither player can write down either that number or any of its divisors.
- Then Bob writes another number, after which again neither player can write down that number or any of its divisors.
- Alice and Bob continue to take turns to write a new number no multiple of which has been written before.
- Whoever manages to play last is the winner.
And As we all know Alice and Bob, they are smart and play the optimal strategy.
Let say the top limit number was $5$:
Alice: $4$ (Then $1$,$2$ and $4$ would be eliminated)
$ - -3- -5$
$ - -3- - $
Who will win at the end, with best play on both sides, if the top-limit is $2017$?