BALTIC-WAY 2012 Problem 10

Two players $A$ and $B$ play the following game. Before the game starts, $A$ chooses 1000 not necessarily different odd primes, and then $B$ chooses half of them and writes them on a blackboard. In each turn a player chooses a positive integer $n$, erases some primes $p_{1}, p_{2}, \ldots, p_{n}$ from the blackboard and writes all the prime factors of $p_{1} p_{2} \ldots p_{n}-2$ instead (if a prime occurs several times in the prime factorization of $p_{1} p_{2} \ldots p_{n}-2$, it is written as many times as it occurs). Player $A$ starts, and the player whose move leaves the blackboard empty loses the game. Prove that one of the two players has a winning strategy and determine who.

Remark: Since 1 has no prime factors, erasing a single 3 is a legal move.