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Chinese remainder theorem

The Chinese remainder theorem is a result about congruences in number theory and its generalizations in abstract algebra. It was first published in the 3rd to 5th centuries by the Chinese mathematician Sun Tzu.
In its basic form, the Chinese remainder theorem will determine a number ''n'' that, when divided by some given divisors, leaves given remainders. For example, what is the lowest number ''n'' that when divided by 3 leaves a remainder of 2, when divided by 5 leaves a remainder of 3, and when divided by 7 leaves a remainder of 2?
== Theorem statement ==
The original form of the theorem, which is contained in the 5th-century book ''Sunzi's Mathematical Classic'' () by the Chinese mathematician Sun Tzu and later generalized with a complete solution called ''Dayanshu'' () in Qin Jiushao's 1247 ''Mathematical Treatise in Nine Sections'' (, ''Shushu Jiuzhang''), is a statement about simultaneous congruences.
Suppose are positive integers that are pairwise coprime. Then, for any given sequence of integers , there exists an integer solving the following system of simultaneous congruences.
:$\begin x \equiv a_1 & \pmod \\ \quad \cdots \\ x \equiv a_k &\pmod \end$
Furthermore, all solutions of this system are congruent modulo the product, . Hence
:$x \equiv y \pmod, \quad 1 \leq i \leq k \qquad \Longleftrightarrow \qquad x \equiv y \pmod.$
Sometimes, the simultaneous congruences can be solved even if the are not pairwise coprime. A solution exists if and only if:
:$a_i \equiv a_j \pmod \qquad \texti\textj$
All solutions are then congruent modulo the least common multiple of the .
Sun Tzu's work contains neither a proof nor a full algorithm. What amounts to an algorithm for solving this problem was described by Aryabhata (6th century; see ). Special cases of the Chinese remainder theorem were also known to Brahmagupta (7th century), and appear in Fibonacci's Liber Abaci (1202).
A modern restatement of the theorem in algebraic language is that for a positive integer with prime factorization
:$n = p_1^\cdots p_k^$
we have the isomorphism between a ring and the direct product of its prime power parts:
:$\mathbf/n\mathbf \cong \mathbf/p_1^\mathbf \times \cdots \times \mathbf/p_k^\mathbf$
The theorem can also be restated in the language of combinatorics as the fact that the infinite arithmetic progressions of integers form a Helly family .

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