Variations on a Theme of Euclid

The game of Euclid is an impartial game played between two players. A position in the game is a pair of integers (a, b). A move consists of replacing the current position with one in which the larger of a and b has been reduced by any multiple of the smaller. The game ends when the two numbers are equal. The players alternate moves, and the winner is the last player to make a move.