# Who has the winning strategy for the game?

My friend Mickey asked me to play the following game with him.

The number $$1$$ is written on the whiteboard. Me and Mickey take turns to do the following, starting with me.

If the number written on the board is $$n$$, then replace it by $$n+d\le2019$$ where $$d|n$$. The player who can not replace the number written on the whiteboard losses.

But I always lose. Actually, does Mickey has a winning strategy?

Any help is appreciated!

Note that if $$n$$ is odd then $$d$$ is odd and $$n+d$$ is always even. On the other hand, if $$n$$ is even then $$1$$ is a divisor of $$n$$ and $$n+1$$ is odd. Therefore it should follow that $$n$$ is a winning position if and only if $$n$$ is even. Starting from $$1$$, the second player has always a winning strategy: replace the current (even) number $$n$$ with $$n+1$$.