Solution:

We prove that Alice wins the game. For argument's sake, suppose that Bob can win by proper play regardless of what Alice does on each of her moves. Note that Alice can force the line $2 \rightarrow \mathbf{4} \rightarrow 6 \rightarrow \mathbf{8} \rightarrow 10 \rightarrow \mathbf{12}$ at the beginning stages of the game. (As each intermediate 'position' from which Bob has to play is a prime power.) Thus the player on turn when the number $12$ is written on the blackboard must be in a 'winning position', i.e., can win the game with skillful play. However, Alice can place herself in that position through the following line that is once again forced for Bob: $2 \rightarrow \mathbf{4} \rightarrow 6 \rightarrow \mathbf{9} \rightarrow 12$. (This time she is in turn with $12$ written on the blackboard.) The obtained contradiction proves our point.