Chinese remainder theorem

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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.

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