If $x_0$ is the solution of the system of linear congruence equations: $$ x \equiv c_1 \text{ mod } m_1$$ $$ x \equiv c_2 \text{ mod } m_2$$ $$ \cdots $$ $$ x \equiv c_s \text{ mod } m_s$$
AND the $\gcd(m_i, m_j) = 1 \forall i \neq j$.
Show that the set of solutions is: $X = \{x_0 + k \cdot m_1 \cdot m_2 \cdot \cdots \cdot m_s \text{ with } k\in \mathbb{Z}\} $.
It makes totally sense to me...but how can I show it mathematically correct? Has anyone a hint for me?