Pass 300Logic Level 5
Let \( k \) be a positive integer. Dan and Sam play a game in which the first to start says the number \(k\) and the one who's next must say a multiple of the previous number, that is between the previous number and its square. They cannot repeat a number even if the number was said by the other. Also, the said number cannot be greater than 300.
For example, Dan begins saying \(3=k\), then Sam can reply 6 or 9, but not 3, because Dan said it before.
The winner is the one who cannot say a number in his turn. If Dan begins, and both players play optimally, for how many numbers \(k\le 200\) does Dan win?