Let b_1, \ldots, b_n \in \mathbb{N}. For $x, y \in \mathbb{Z}^n$, define $x \cdot y$ as $\newcommand{\lcm}{\operatorname{lcm}}$ $x \cdot y = \left(\sum_{i=1}^n (x_i \text{ mod } b_i)(y_i \text{ mod } b_i) \text{ mod } b_i\right) \text{ mod } \lcm(b_1, \ldots, b_n)$
Informally, each multiplication $x_iy_i$ is carried over $\mathbb{Z}/b_i\mathbb{Z}$ and the sum is carried over $\mathbb{Z}/ \lcm(b_1, \ldots, b_n)\mathbb{Z}$.
My question is the following: does x \cdot (y + z) \equiv x \cdot y + x \cdot z \quad (\text{mod} \lcm(b_1, \ldots, b_n)) hold?