Alice and Bob take turns playing a game, with Alice starting first.
Initially, there is a number N on the chalkboard. On each player's turn, that player makes a move consisting of:
- Choosing any x with 0 < x < N and N % x == 0.
- Replacing the number N on the chalkboard with N - x.
Also, if a player cannot make a move, they lose the game.
Return True if and only if Alice wins the game, assuming both players play optimally.
Example 1:
Input: 2
Output: true
Explanation: Alice chooses 1, and Bob has no more moves.
Example 2:
Input: 3
Output: false
Explanation: Alice chooses 1, Bob chooses 1, and Alice has no more moves.
Note:
1 <= N <= 1000