Let $a, b, c, p, q, r$ be positive integers with $p, q, r \ge 2$. Denote
$$Q = \{(x, y, z) \in \mathbb{Z}^3 : 0 \le x \le a, 0 \le y \le b, 0 \le z \le c\}.$$
Initially, some pieces are placed on each point in $Q$, with a total of $M$ pieces. The following three types of operations can be performed repeatedly:
1. Remove $p$ pieces from $(x, y, z)$ and place a piece on $(x-1, y, z)$;
2. Remove $q$ pieces from $(x, y, z)$ and place a piece on $(x, y-1, z)$;
3. Remove $r$ pieces from $(x, y, z)$ and place a piece on $(x, y, z-1)$.

We need to find the smallest positive integer $M$ such that it is always possible to perform a sequence of operations to place a piece on $(0,0,0)$, regardless of the initial distribution of pieces.

We claim that the smallest positive integer $M$ is $p^a q^b r^c$.

To show that $M \ge p^a q^b r^c$, consider the invariant
$$N = \sum_{0 \le x \le a, 0 \le y \le b, 0 \le z \le c} f(x, y, z) p^{-x} q^{-y} r^{-z},$$
where $f(x, y, z)$ is the number of pieces at $(x, y, z)$. Initially, $N = M p^{-a} q^{-b} r^{-c}$. After any operation, $N$ remains unchanged. To ensure that $f(0,0,0) \ge 1$ at the end, we need $N \ge 1$, which implies $M \ge p^a q^b r^c$.

To show that $M = p^a q^b r^c$ is sufficient, we proceed by induction on $a + b + c$. The base case is trivial. Assume it holds for all smaller values of $a + b + c$. We group the $p^a q^b r^c$ pieces into $p$ groups of $p^{a-1} q^b r^c$ pieces each. By the inductive hypothesis, each group can be used to place a piece on $(1, 0, 0)$. These $p$ pieces can then be used to place a piece on $(0, 0, 0)$.

Thus, the smallest positive integer $M$ such that a piece can always be placed on $(0,0,0)$ is:
$$\boxed{p^a q^b r^c}.$$