Strategy of a game

If at any time Bob chooses $1$ then Alice chooses $97$ and wins. So in the following I will ignore this choice for Bob.

Alice wins by playing $62$. There are two possibilities.

Bob chooses $31$

Alice now forces the following sequence where Alice's choices are asterisked. She always chooses a product of two primes greater than $50$ where one of the primes has already been used and so Bob has no choice (other than $1$).

$*93*,3,*51*,17,*85*,5,*95*,19,*57*$ and now $1$ is forced.

Bob chooses $2$

Alice now forces the following sequence.

$*58*,29,*87*,3,*51*$ and now as in the above sequence.