JBMO-SL 2021 N4

Dragos, the early ruler of Moldavia, and Maria the Oracle play the following game. Firstly, Maria chooses a set $S$ of prime numbers. Then Dragos gives an infinite sequence $x_1, x_2, ...$ of distinct positive integers. Then Maria picks a positive integer $M$ and a prime number $p$ from her set $S$ . Finally, Dragos picks a positive integer $N$ and the game ends. Dragos wins if and only if for all integers $n \ge N$ the number $x_n$ is divisible by $p^M$ ; otherwise, Maria wins. Who has a winning strategy if the set S must be: $\hspace{5px}$ a) finite; $\hspace{5px}$ b) infinite?

Proposed by Boris Stanković, Bosnia and Herzegovina