Given a fixed integer n = 2, let S be the set of all factors of n. Ellie and Neil play a game where they take turns removing elements from S with Ellie going first. After the first turn, a number removed on each turn must be relatively prime to the number removed in the previous turn by the opponent. The first player who is unable to remove a number on their turn (including starting a turn with S empty) loses. What is the smallest composite n that Neil has a winning strategy on?

Question

Given a fixed integer n >= 2, let S be the set of all factors of n. Ellie and Neil play a game where they take turns removing elements from S with Ellie going first. After the first turn, a number removed on each turn must be relatively prime to the number removed in the previous turn by the opponent. The first player who is unable to remove a number on their turn (including starting a turn with S empty) loses. What is the smallest composite n that Neil has a winning strategy on?

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Rules of the Game: To find the smallest composite number n for which Neil has a winning strategy, we need to consider the rules of the game. The first player (Ellie) will remove a factor of n, and then Neil must remove a factor that is relatively prime to the factor Ellie removed. If Neil can always choose a factor that is relatively prime to Ellie's choice, he will win.n = 4: Let's start with the smallest composite number, which is 4. The set of factors for 4 is S = {1, 2, 4}. Ellie can start by removing any factor. If she removes 1, Neil can remove 2 or 4, but both are not relatively prime to 1. If she removes 2, Neil can only remove 1, which is not relatively prime to 2. If she removes 4, Neil can only remove 1, which is not relatively prime to 4. Therefore, Neil does not have a winning strategy for n = 4.n = 6: Next, let's consider n = 6, which has the set of factors S = {1, 2, 3, 6}. If Ellie removes 1, Neil can remove 5, which is not a factor of 6, so this is not a valid move. If Ellie removes 2, Neil can remove 3, which is relatively prime to 2. If Ellie removes 3, Neil can remove 2, which is relatively prime to 3. If Ellie removes 6, Neil can remove 5, which is not a factor of 6, so this is not a valid move. Therefore, Neil does not have a winning strategy for n = 6 because he cannot always make a valid move that is relatively prime to Ellie's choice.n = 8: Now, let's consider n = 8, which has the set of factors S = {1, 2, 4, 8}. If Ellie removes 1, Neil cannot make a valid move because all remaining factors are not relatively prime to 1. If Ellie removes 2, Neil can remove 1, which is not relatively prime to 2. If Ellie removes 4, Neil can remove 1, which is not relatively prime to 4. If Ellie removes 8, Neil can remove 1, which is not relatively prime to 8. Therefore, Neil does not have a winning strategy for n = 8.n = 9: Next, let's consider n = 9, which has the set of factors S = {1, 3, 9}. If Ellie removes 1, Neil cannot make a valid move because all remaining factors are not relatively prime to 1. If Ellie removes 3, Neil can remove 1, which is not relatively prime to 3. If Ellie removes 9, Neil can remove 1, which is not relatively prime to 9. Therefore, Neil does not have a winning strategy for n = 9.n = 10: Now, let's consider n = 10, which has the set of factors S = {1, 2, 5, 10}. If Ellie removes 1, Neil cannot make a valid move because all remaining factors are not relatively prime to 1. If Ellie removes 2, Neil can remove 5, which is relatively prime to 2. If Ellie removes 5, Neil can remove 2, which is relatively prime to 5. If Ellie removes 10, Neil can remove 3, which is not a factor of 10, so this is not a valid move. Therefore, Neil does not have a winning strategy for n = 10.n = 12: Let's consider n = 12, which has the set of factors S = {1, 2, 3, 4, 6, 12}. If Ellie removes 1, Neil can remove 5, which is not a factor of 12, so this is not a valid move. If Ellie removes 2, Neil can remove 3 or 5, which are relatively prime to 2, but 5 is not a factor of 12, so he must choose 3. If Ellie removes 3, Neil can remove 2 or 5, which are relatively prime to 3, but again, 5 is not a factor of 12, so he must choose 2. If Ellie removes 4, Neil can remove 3, which is relatively prime to 4. If Ellie removes 6, Neil can remove 5, which is not a factor of 12, so this is not a valid move. If Ellie removes 12, Neil can remove 5, which is not a factor of 12, so this is not a valid move. Therefore, Neil does not have a winning strategy for n = 12.n = 15: Finally, let's consider n = 15, which has the set of factors S = {1, 3, 5, 15}. If Ellie removes 1, Neil can remove 2, which is not a factor of 15, so this is not a valid move. If Ellie removes 3, Neil can remove 5, which is relatively prime to 3. If Ellie removes 5, Neil can remove 3, which is relatively prime to 5. If Ellie removes 15, Neil can remove 2, which is not a factor of 15, so this is not a valid move. However, after Neil removes 3 or 5, Ellie cannot make a move because the remaining factors are not relatively prime to Neil's choice. Therefore, Neil has a winning strategy for n = 15.

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