Editing Chinese remainder theorem (section)

==Statement==
Let ''n''<sub>1</sub>, ..., ''n''<sub>''k''</sub> be integers greater than 1, which are often called ''[[modular arithmetic|moduli]]'' or ''[[Euclidean division|divisors]]''. Let us denote by ''N'' the product of the ''n''<sub>''i''</sub>.

The Chinese remainder theorem asserts that if the ''n''<sub>''i''</sub> are [[pairwise coprime]], and if ''a''<sub>1</sub>, ..., ''a''<sub>''k''</sub> are integers such that 0 ≤ ''a''<sub>''i''</sub> < ''n''<sub>''i''</sub> for every ''i'', then there is one and only one integer ''x'', such that 0 ≤ ''x'' < ''N'' and the remainder of the [[Euclidean division]] of ''x'' by ''n''<sub>''i''</sub> is ''a''<sub>''i''</sub> for every ''i''.

This may be restated as follows in terms of [[modular arithmetic|congruences]]:
If the <math>n_i</math> are pairwise coprime, and if ''a''<sub>1</sub>, ..., ''a''<sub>''k''</sub> are any integers, then the system

:<math>\begin{align}
 x &\equiv a_1  \pmod{n_1} \\
  &\,\,\,\vdots \\
 x &\equiv a_k \pmod{n_k},
\end{align}</math>
has a solution, and any two solutions, say ''x''<sub>1</sub> and ''x''<sub>2</sub>, are congruent modulo ''N'', that is, {{math|''x''<sub>1</sub> ≡ ''x''<sub>2</sub> (mod ''N''{{hairsp}})}}.<ref>{{harvnb|Ireland|Rosen|1990|page=34}}</ref>

In [[abstract algebra]], the theorem is often restated as: if the ''n''<sub>''i''</sub> are pairwise coprime, the map
:<math>x \bmod N \;\mapsto\;(x \bmod n_1,\, \ldots,\, x \bmod n_k)</math>
defines a [[ring isomorphism]]<ref>{{harvnb|Ireland|Rosen|1990|page=35}}</ref>

:<math>\mathbb{Z}/N\mathbb{Z} \cong \mathbb{Z}/n_1\mathbb{Z} \times \cdots \times \mathbb{Z}/n_k\mathbb{Z}</math>
between the [[ring (mathematics)|ring]] of [[integers modulo n|integers modulo ''N'']] and the [[product of rings|direct product]] of the rings of integers modulo the ''n''<sub>''i''</sub>. This means that for doing a sequence of arithmetic operations in <math>\mathbb{Z}/N\mathbb{Z},</math> one may do the same computation independently in each <math>\mathbb{Z}/n_i\mathbb{Z}</math> and then get the result by applying the isomorphism (from the right to the left). This may be much faster than the direct computation if ''N'' and the number of operations are large. This is widely used, under the name ''multi-modular computation'', for [[linear algebra]] over the integers or the [[rational number]]s.

The theorem can also be restated in the language of [[combinatorics]] as the fact that the infinite [[arithmetic progression]]s of integers form a [[Helly family]].<ref>{{harvnb|Duchet|1995}}</ref>