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Chinese Remainder Theorem

Find integers x that leave remainders 2, 3, 2 when divided by 3, 5, 7, respectively. [Sun-Tsu, 100 A.D.]

Theorem 31.27 Let \(n \;=\; n_1 n_2 \cdots n_k\), where ni are pairwise relatively prime and consider the correspondence

\begin{displaymath}a \;\leftrightarrow\; (a_1, a_2, \ldots, a_k),\end{displaymath}

where \(a \in Z_n, \;a_i \in Z_{n_i}\), and ai = a mod nifor i = 1, $\cdots$,k.


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