Summary

The results of this paper can be summarized as follows:

• Player 1 trying for a common factor
  • If $N$ is even, Player 2 wins
  • If $N$ is odd, and $N \geq 11$, Player 1 wins
  • If $N$ is odd, and $N_{0}$ is even, Player 1 wins.
  • If $N$ is odd, $N \geq 7$, and $N_{0} \equiv 3 \bmod 6$, Player 1 wins.
  • If $N$ is odd, $N \geq 9$, and $N_{0} \equiv 1 \bmod 6$, Player 1 wins.
  • In all other cases, Player 2 wins
• Player 2 trying for a common factor
  • If $N$ is odd, Player 1 wins
  • If $N$ is even, and $N \geq 12$, Player 2 wins
  • If $N$ is even, and $N \geq 10$, and $N_{0} \equiv 5 \bmod 6$, or $N_{0} \equiv 1 \bmod 6$, or $N_{0} \equiv 0 \bmod 6$, Player 2 wins
  • If $N$ is even, and $N \geq 8$, and $N_{0} \equiv 3 \bmod 6$, or $N_{0} \equiv 2 \bmod 6$, Player 2 wins
  • In all other cases, Player 1 wins.