# Divisor game

Let $$N \ge 2$$ be a positive integer. Alice and Bob play the following divisor game: starting with the set $$D_N$$ of positive divisors of $$N$$, the players take turns removing some elements from the set. At a player's turn, he or she chooses a divisor $$d$$ that remains, and removes $$d$$ and any of the divisors of $$d$$ that remain. The player who moves last loses.

For example: $$N = 18, D_N = \{ 1,2,3,6,9,18 \}$$. Alice chooses $$d=2$$, so she removes $$1$$ and $$2$$. The remaining set is $$\{3,6,9,18\}$$. Bob chooses $$d = 9$$ and so must also remove $$3$$. The remaining set is $$\{6,18\}$$. Alice takes $$6$$, and now Bob is forced to take $$18$$, so he loses.

Let $$n \ge 2$$ be the largest positive integer $$\le 200$$ such that if Alice and Bob play the divisor game for $$n$$ and both of them play optimally, the second player wins. Find $$n$$. If no such $$n$$ exists, enter $$999$$.

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