Abstract Euclidis a well-known two-player impartial combinatorial game. A position in Euclidis a pair of positive integers and the players move alternately by subtracting a positive integer multiple of one of the integers from the other integer without making the result negative. The player who makes the last move wins. There is a variation of Eucliddue to Grossman in which the game stops when the two entries are equal. We examine a further variation which we called M-Euclidwhere the game stops when one of the entries is a positive integer multiple of the other. We solve the Sprague–Grundy function for M-Euclidand compare the Sprague–Grundy functions of the three games.