Alphonse and Beryl play the following game. Two positive integers m and n are written on the board. On each turn, a player selects one of the numbers on the board, erases it, and writes in its place any positive divisor of this number as long as it is different from any of the numbers previously written on the board. For example, if 10 and 17 are written on the board, a player can erase 10 and write 2 in its place (as long as 2 has not appeared on the board before). The player who cannot make a move loses. Alphonse goes first.
(a) Suppose m=2^{40} and n=3^{51}. Determine which player is always able to win the game and explain the winning strategy.
(b) Suppose m=2^{40} and n=2^{51}. Determine which player is always able to win the game and explain the winning strategy.