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 [i]Boris Stanković, Bosnia and Herzegovina[/i]
Let's analyze the problem in two parts: when the set \( S \) is finite and when \( S \) is infinite.

### Part (a): \( S \) is finite

1. **Define the set \( S \)**:
   Let \( S = \{ p_1, p_2, \ldots, p_k \} \) be a finite set of prime numbers.

2. **Construct the sequence \( \{x_n\} \)**:
   Define the sequence \( \{x_n\} \) as follows:
   \[
   x_n = (p_1 p_2 \cdots p_k)^n
   \]
   This means each \( x_n \) is the product of all primes in \( S \) raised to the power \( n \).

3. **Maria's choice**:
   Maria picks a positive integer \( M \) and a prime number \( p \) from her set \( S \).

4. **Dragos's strategy**:
   Dragos needs to ensure that for all \( n \ge N \), \( x_n \) is divisible by \( p^M \). Since \( x_n = (p_1 p_2 \cdots p_k)^n \), each \( x_n \) is divisible by \( p^n \) for any \( p \in S \).

5. **Conclusion**:
   Since \( n \ge N \) and \( n \) can be made arbitrarily large, \( p^M \) will always divide \( p^n \) for sufficiently large \( n \). Therefore, Dragos can always choose \( N \) such that \( n \ge N \) ensures \( x_n \) is divisible by \( p^M \).

Thus, Dragos has a winning strategy when \( S \) is finite.

### Part (b): \( S \) is infinite

1. **Define the set \( S \)**:
   Let \( S = \{ p_1, p_2, \ldots \} \) be an infinite set of prime numbers, where \( p_1 < p_2 < \cdots \).

2. **Construct the sequence \( \{x_n\} \)**:
   Define the sequence \( \{x_n\} \) as follows:
   \[
   x_n = (p_1 p_2 \cdots p_n)^n
   \]
   This means each \( x_n \) is the product of the first \( n \) primes in \( S \) raised to the power \( n \).

3. **Maria's choice**:
   Maria picks a positive integer \( M \) and a prime number \( p \) from her set \( S \).

4. **Dragos's strategy**:
   Dragos needs to ensure that for all \( n \ge N \), \( x_n \) is divisible by \( p^M \). Since \( x_n = (p_1 p_2 \cdots p_n)^n \), each \( x_n \) is divisible by \( p^n \) for any \( p \in \{ p_1, p_2, \ldots, p_n \} \).

5. **Conclusion**:
   Since \( S \) is infinite, for any prime \( p \) chosen by Maria, there exists an \( n \) such that \( p \) is among the first \( n \) primes. Therefore, \( x_n \) will be divisible by \( p^n \), and for sufficiently large \( n \), \( p^M \) will divide \( p^n \).

Thus, Dragos has a winning strategy when \( S \) is infinite.

The final answer is Dragos has a winning strategy in both cases: when \( \boxed{ S } \) is finite and when \( S \) is infinite.