For questions related to the Chinese Remainder Theorem and its applications.
In number theory and algebra, the Chinese Remainder Theorem states that if $n_1, ..., n_k$ are positive, pairwise relatively prime integers, and $a_1, ..., a_k$ is any collection of integers, there is a unique solution to the system of congruences
$$x \equiv a_1 \pmod{n_1}$$ $$x \equiv a_2 \pmod{n_2}$$ $$\vdots$$ $$x \equiv a_k \pmod{n_k}$$
modulo the product $n_1 n_2 \cdots n_k$.
More generally, consider a principal ideal domain $R$. If $u_1, ..., u_k$ are pairwise relatively prime elements of $R$ and $u = u_1 \cdots u_k$, then there is an isomorphism
$$R/uR \cong R/u_1 R \times \cdots \times R/u_k R$$
by the map
$$x + uR \mapsto (x + u_1 R, ..., x + u_k R)$$
Reference: Chinese remainder theorem.
