Question 22
Given positive integers $n$, $r$ and distinct prime numbers $p_1$, $p_2$, $\dots$, $p_r$. Initially, there are $(n+1)^r$ numbers on the blackboard: $p_1^{i_1} p_2^{i_2} \cdots p_r^{i_r}$ ($0 \le i_1, i_2, \dots, i_r \le n$). Alice and Bob take turns (Alice goes first) to make the following moves, until only one number is left on the blackboard:
* Every time, Alice erases two numbers (can be identical) and writes their greatest common divisor on the blackboard.
* Every time, Bob erases two numbers (can be identical) and writes their least common multiple on the blackboard.
Find the least integer $M$, such that Alice can guarantee the remaining number does not exceed $M$.
* Every time, Alice erases two numbers (can be identical) and writes their greatest common divisor on the blackboard.
* Every time, Bob erases two numbers (can be identical) and writes their least common multiple on the blackboard.
Find the least integer $M$, such that Alice can guarantee the remaining number does not exceed $M$.