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Chinese remainder theorem (Theorem)

Suppose we have a set of n congruences of the form

where p1 p2 pn are relatively prime. Let

P= ni=1pi and, for all i (1 i n ), let yi be an integer that satisfies

yi Ppi 1(modpi) Then one solution of these congruences is

x0= ni=1aiyi Ppi Any x satisfies the set of congruences if and only if it satisfies

x x0(modP)

The Chinese remainder theorem originated in the book ``Sun Zi Suan Jing'', or Sun Tzu's Arithmetic Classic, by the Chinese mathematician Sun Zi, or Sun Tzu, who also wrote ``Sun Zi Bing Fa'', or Sun Tzu's The Art of War. The theorem is said to have been used to count the size of the ancient Chinese armies (i.e., the soldiers would split into groups of 3, then 5, then 7, etc, and the ``leftover'' soldiers from each grouping would be counted).

"Chinese remainder theorem" is owned by CWoo. [ full author list (4) | owner history (4) ]

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Source URL: http://www.planetmath.org/encyclopedia/ChineseRemainderTheorem.html Saylor URL: http://www.saylor.org/courses/cs409 Attributed to [Chi Woo]

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See Also: Chinese remainder theorem in terms of divisor theory, Gödel's beta function

Attachments: Chinese remainder theorem proof (Proof) by vampyr noncommutative case Chinese remainder theorem for rings (Theorem) by polarbear

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Cross-references: groups, size, theorem, arithmetic, solution, integer, relatively prime, congruences There are 15 references to this entry. This is version 9 of Chinese remainder theorem, born on 2001-11-16, modified 2008-06-13. Object id is 879, canonical name is ChineseRemainderTheorem. Accessed 26136 times total. Classification: AMS MSC: 11D79 (Number theory :: Diophantine equations :: Congruences in many variables)

 

Source URL: http://www.planetmath.org/encyclopedia/ChineseRemainderTheorem.html Saylor URL: http://www.saylor.org/courses/cs409 Attributed to [Chi Woo]

Saylor.org Page 2 of 2