Problem - 2801
Two players, $A$ and $B$, take turns naming positive integers, with $A$ playing first. No player may name an integer that can be expressed as a linear combination, with positive integer coefficients, of previously named integers. The player who names 1 loses. Show that no matter how A and B play, the game will always end.
This is the game of Sylver Coinage which is invented by John H. Conway. It is named after James Joseph Sylvester who proved that if $a$ and $b$ are relatively prime positive integers, then the largest positive integer that cannot be expressed as a positive linear combination of $a$ and $b$ is $(a - 1)(b - 1) - 1$.