Question 182
Let $a, b, c, p, q, r$ be positive integers with $p, q, r \ge 2$. Denote
$$
Q = \{(x, y, z) \in \mathbb{Z}^3 \mid 0 \le x \le a, 0 \le y \le b, 0 \le z \le c\}.
$$
Initially, some stones are put at each point of $Q$, with total $M$ stones. Then one can perform the following three types of operations repeatedly:
(1) remove $p$ stones on $(x, y, z)$ and place a stone on $(x-1, y, z)$;
(2) remove $q$ stones on $(x, y, z)$ and place a stone on $(x, y-1, z)$;
(3) remove $r$ stones on $(x, y, z)$ and place a stone on $(x, y, z-1)$.
Find the smallest positive integer $M$ such that one can always perform a sequence of operations, to place a stone at $(0, 0, 0)$, no matter how the stones were distributed initially.
$$
Q = \{(x, y, z) \in \mathbb{Z}^3 \mid 0 \le x \le a, 0 \le y \le b, 0 \le z \le c\}.
$$
Initially, some stones are put at each point of $Q$, with total $M$ stones. Then one can perform the following three types of operations repeatedly:
(1) remove $p$ stones on $(x, y, z)$ and place a stone on $(x-1, y, z)$;
(2) remove $q$ stones on $(x, y, z)$ and place a stone on $(x, y-1, z)$;
(3) remove $r$ stones on $(x, y, z)$ and place a stone on $(x, y, z-1)$.
Find the smallest positive integer $M$ such that one can always perform a sequence of operations, to place a stone at $(0, 0, 0)$, no matter how the stones were distributed initially.