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 p1, p2, …, pn from the blackboard and writes all the prime factors of p1p2⋯pn−2 instead (if a prime occurs several times in the prime factorization of p1p2⋯pn−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. modular arithmeticnumber theoryprime factorizationcombinatorics unsolvedcombinatorics