Omni-MATH / Let a,b,c,p,q,r be positive integers with p,q,r≥2. Denote

Let a,b,c,p,q,r be positive integers with p,q,r≥2. Denote
Q={(x,y,z)∈Z³:0≤x≤a,0≤y≤b,0≤z≤c}.
Initially, some pieces are put on the each point in Q, with a total of M pieces. Then, one can perform the following three types of operations repeatedly:
(1) Remove p pieces on (x,y,z) and place a piece on (x−1,y,z) ;
(2) Remove q pieces on (x,y,z) and place a piece on (x,y−1,z) ;
(3) Remove r pieces on (x,y,z) and place a piece on (x,y,z−1).

Find the smallest positive integer M such that one can always perform a sequence of operations, making a piece placed on (0,0,0), no matter how the pieces are distributed initially.

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 put on the each point in $Q$, with a total of $M$ pieces. Then, one can perform the following three types of operations repeatedly:
(1) Remove $p$ pieces on $(x, y, z)$ and place a piece on $(x-1, y, z)$ ;
(2) Remove $q$ pieces on $(x, y, z)$ and place a piece on $(x, y-1, z)$ ;
(3) Remove $r$ pieces on $(x, y, z)$ and place a piece on $(x, y, z-1)$.

Find the smallest positive integer $M$ such that one can always perform a sequence of operations, making a piece placed on $(0,0,0)$, no matter how the pieces are distributed initially.