Chinese remainder theorem

Let $LaTeX: m_1, m_2, \ldots, m_k$ be relatively prime positive integers, and let $LaTeX: b_1, b_2, \ldots, b_k$ be any integers. Then, there exists $LaTeX: x$ such that $LaTeX: x = b_i(\hspace*{-8pt}\mod x_i)$ for each $LaTeX: i$. The value of $LaTeX: x$ is uniquely determined by $LaTeX: m_1, m_2, \ldots, m_k$ and $LaTeX: b_1, b_2, \ldots, b_k$.