D. Deleting Divisors
games; math; number theory
Alice and Bob are playing a game. They start with a positive integer \(n\) and take alternating turns doing operations on it. Each turn a player can subtract from \(n\) one of its divisors that isn't \(1\) or \(n\). The player who cannot make a move on his/her turn loses. Alice always moves first.Note that they subtract a divisor of the current number in each turn.You are asked to find out who will win the game if both players play optimally.
The first line contains a single integer \(t\) (\(1 \leq t \leq 10^4\)) — the number of test cases. Then \(t\) test cases follow.Each test case contains a single integer \(n\) (\(1 \leq n \leq 10^9\)) — the initial number.
For each test case output ""Alice"" if Alice will win the game or ""Bob"" if Bob will win, if both players play optimally.
In the first test case, the game ends immediately because Alice cannot make a move.In the second test case, Alice can subtract \(2\) making \(n = 2\), then Bob cannot make a move so Alice wins.In the third test case, Alice can subtract \(3\) so that \(n = 9\). Bob's only move is to subtract \(3\) and make \(n = 6\). Now, Alice can subtract \(3\) again and \(n = 3\). Then Bob cannot make a move, so Alice wins.