Simple Number-Picking Games

This is really a post about two games, one simple, and the other simpler.

The simple game first.

For any plural number of players.

Players take turns. On each turn, a player picks one number from (2,3,4,...,n) that has not yet been picked in the game. (n is, say, 100.)

Each number, after the first pick, must NOT be coprime to the number picked in the previous move by the last player to move.
Or
If the last number picked in the game was a prime AND no multiples of that prime exist among the numbers that have not yet been picked, then the player may pick any prime from the primes that have not yet been picked.

The last player able to move wins.

--

Simpler game:

For 2 players.

As before, players take turns. On each turn, a player picks one number from (2,3,4,...,n) that has not yet been picked in the game.
Each number, after the first pick, must NOT be coprime to the number picked in the previous move by the last player to move.

If a player can move, the player must move.

The last player to move LOSES.

Thanks,
Leroy Quet