**Chinese Remainder Theorem**

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Suppose are positive integers and coprime in pair. For any sequence of integers , there exists an integer x solving the following system of congruence equations:

There exists an unique modulo solution of the system of simultaneous congruences above:

in which:

M &= m_1 \cdots m_k \\

M_1 &= \frac{M}m_1 , \cdots, M_k = \frac{M}m_k \\

y_1 &\equiv (M_1)^{-1} \pmod{m_1}, \cdots , y_k\equiv (M_k)^{-1}\pmod{m_k}

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