# Nim Variant (reducing by divisors)

Alice and Bob play the following game. They choose a number $N$ to play with. The rules are as follows:

1. Alice plays first, and the two players alternate.
2. In his/her turn, a player can subtract from $N$ any proper divisor (not equal to $N$) of $N$. The number thus obtained is the new $N$.
3. The person who cannot make a move in his/her turn loses the game. Assuming both play optimally, who wins the game ?
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• $N=1$ is a losing position as there is no move.
• $N>1$ and even is a winning position: you can subtract an odd proper divisor such as $1$ to reach an odd number
• $N>1$ and odd is a losing position: you must subtract an odd proper divisor to reach an even number