1. Chinese Remainder Theorem - Wikipedia, The Free Encyclopedia The Chinese remainder theorem is a result about congruences in number theory and its generalizations in abstract algebra. http://en.wikipedia.org/wiki/Chinese_remainder_theorem

Chinese remainder theorem From Wikipedia, the free encyclopedia Jump to: navigation search The Chinese remainder theorem is a result about congruences in number theory and its generalizations in abstract algebra Contents Theorem statement A constructive algorithm to find the solution Statement for principal ideal domains Statement for general rings ... edit Theorem statement The original form of the theorem, contained in a third-century AD book Sun Zi suanjing (孫子算經 The Mathematical Classic by Sun Zi) by Chinese mathematician Sun Tzu and later republished in a 1247 book by Qin Jiushao , the Shushu Jiuzhang Mathematical Treatise in Nine Sections ) is a statement about simultaneous congruences (see modular arithmetic Suppose n n n k are positive integers which are pairwise coprime . Then, for any given set of integers a a a k , there exists an integer x solving the system of simultaneous congruences Furthermore, all solutions x to this system are congruent modulo the product N n n n k Hence for all , if and only if Sometimes, the simultaneous congruences can be solved even if the n i s are not pairwise coprime. A solution

2. Chinese Remainder Theorem Summary And Analysis Summary | BookRags.com Chinese remainder theorem summary with 6 pages of lesson plans, quotes, chapter summaries, analysis, encyclopedia entries, essays, research information, and more. http://www.bookrags.com/Chinese_remainder_theorem

3. Talk:Chinese Remainder Theorem - Wikipedia, The Free Encyclopedia You can t always perform the Euclidean Algorithm in principal ideal domains http://en.wikipedia.org/wiki/Talk:Chinese_remainder_theorem

Talk:Chinese remainder theorem From Wikipedia, the free encyclopedia Jump to: navigation search Mathematics portal This article is within the scope of , a collaborative effort to improve the coverage of Mathematics on Wikipedia. If you would like to participate, please visit the project page , where you can join the discussion and see a list of open tasks. Start High Priority Field Number theory Please update this rating as the article progresses, or if the rating is inaccurate. Click to show/hide comments. Please add to or update the comments to suggest improvements to the article. Longer lead, and more explanation would help. Geometry guy 14:18, 21 May 2007 (UTC) Contents Domains Versions for more than two numbers Euclidean algorithm More practical approach ... edit Domains I wonder how general this theorem is true? It works for principal ideal domains, but is it still true for unique factorization domains? AxelBoldt edit Versions for more than two numbers I'd like to say that the result is true for three or more numbers in the place of a and b (pairwise coprime), but I'm having difficulty phrasing this without making it seem more complicated than it is. Matthew Woodcraft I added the versions for more than two factors. The description of how to solve the congruencies and of the inverse isomorphisms are still missing. AxelBoldt

4. Chinese Remainder Theorem - Wikipedia@pedia Chinese remainder theoremThe Chinese remainder theorem is a result about congruences in number theory and its generalizations in abstract algebra.Contents1 Theorem statement2 A http://wikipedia.atpedia.com/en/articles/c/h/i/Chinese_remainder_theorem.html

wikipedia＠pedia wikipedia@PEDIA is study site of the language based on Wikipedia. TOP Translation Select text and it is translated. to AFRIKAANS to ALBANIAN to AMHARIC to ARABIC to ARMENIAN to AZERBAIJANI to BASQUE to BELARUSIAN to BENGALI to BIHARI to BULGARIAN to BURMESE to CATALAN to CHEROKEE to CHINESE to CROATIAN to CZECH to DANISH to DHIVEHI to DUTCH to ENGLISH to ESPERANTO to ESTONIAN to FILIPINO to FINNISH to FRENCH to GALICIAN to GEORGIAN to GERMAN to GREEK to GUARANI to GUJARATI to HEBREW to HINDI to HUNGARIAN to ICELANDIC to INDONESIAN to INUKTITUT to ITALIAN to JAPANESE to KANNADA to KAZAKH to KHMER to KOREAN to KURDISH to KYRGYZ to LAOTHIAN to LATVIAN to LITHUANIAN to MACEDONIAN to MALAY to MALAYALAM to MALTESE to MARATHI to MONGOLIAN to NEPALI to NORWEGIAN to ORIYA to PASHTO to PERSIAN to POLISH to PORTUGUESE to PUNJABI to ROMANIAN to RUSSIAN to SANSKRIT to SERBIAN to SINDHI to SINHALESE to SLOVAK to SLOVENIAN to SPANISH to SWAHILI to SWEDISH to TAJIK to TAMIL to TAGALOG to TELUGU to THAI to TIBETAN to TURKISH to UKRAINIAN to URDU to UZBEK to UIGHUR to VIETNAMESE This area is result which is translated word.

5. Chinese Remainder Theorem Chinese Remainder Theorem from WN Network. WorldNews delivers latest Breaking news including World News, US, politics, business, entertainment, science, http://wn.com/Chinese_remainder_theorem

6. Chinese Remainder Theorem - AoPSWiki Jul 19, 2008 The Chinese Remainder Theorem is a number theoretic result. It is one of the only theorems named for an oriental person or place, http://www.artofproblemsolving.com/Wiki/index.php/Chinese_Remainder_Theorem

Art of Problem Solving LOGIN/REGISTER Home Bookstore AoPS Curriculum ... Articles You are here: Resources AoPSWiki Search Wiki Page Tools Page Discussion View source History Personal tools Talk for this IP Log in Navigation Main Page Community portal Current events Recent changes ... Help Toolbox What links here Related changes Special pages Printable version ... Permanent link Chinese Remainder Theorem From AoPSWiki The Chinese Remainder Theorem is a number theoretic result. It is one of the only theorems named for an oriental person or place, due to the closed development of mathematics in the western world. Contents Theorem Proof Applicability Extended version of the theorem ... Discussion Theorem Let be relatively prime to . Then each residue class mod is equal to the intersection of a unique residue class mod and a unique residue class mod , and the intersection of each residue class mod with a residue class mod is a residue class mod Proof If , then and clearly differ by a multiple of , so and . This is the first part of the theorem. The converse follows because and must differ by a multiple of and , and . This is the second part of the theorem. Applicability Much like the Fundamental Theorem of Arithmetic , many people seem to take this theorem for granted before they consciously turn their attention to it. It ubiquity derives from the fact that many results can be easily proven mod (a power of a prime), and can then be generalized to mod

7. Chinese Remainder Theorem For example, if a1 = 3 and a2 = 5, the Chinese remainder theorem (CRT) says that every integer from 0 to 14 will have a unique set of remainders when http://www.daviddarling.info/encyclopedia/C/Chinese_remainder_theorem.html

NUMBER THEORY A B C ... CONTACT entire Web this site Chinese remainder theorem If there are n numbers, a to a n , that have no factors in common (in other words, are pairwise relatively prime), then any integer greater than or equal to and less than the product of all the numbers n can be uniquely represented by a series consisting of the remainders of division by the numbers n . For example, if a = 3 and a = 5, the Chinese remainder theorem (CRT) says that every integer from to 14 will have a unique set of remainders when divided separately by ( modulo ) 3 and 5. Listing out all the possibilities shows that this is true: has a remainder of modulo 3 and a remainder of modulo 5. 1 has a remainder of 1 modulo 3 and a remainder of 1 modulo 5. 2 has a remainder of 2 modulo 3 and a remainder of 2 modulo 5. 3 has a remainder of modulo 3 and a remainder of 3 modulo 5. 4 has a remainder of 1 modulo 3 and a remainder of 4 modulo 5. 5 has a remainder of 2 modulo 3 and a remainder of modulo 5.

8. Chinese Remainder Theorem - Encyclopedia Article - Citizendium This is a draft article, under development and not meant to be cited; you can help to improve it. These unapproved articles are subject to a disclaimer. http://locke.citizendium.org:8080/wiki/Chinese_remainder_theorem

Chinese remainder theorem From Citizendium, the Citizens' Compendium Jump to: navigation search addthis_pub = 'citizendium'; addthis_logo = ''; addthis_logo_color = ''; addthis_logo_background = ''; addthis_brand = 'Citizendium'; addthis_options = ''; addthis_offset_top = ''; addthis_offset_left = ''; Main Article Talk Related Articles Bibliography External Links ... Advanced This is a draft article , under development and not meant to be cited; you can help to improve it. These unapproved articles are subject to edit intro The Chinese remainder theorem is a mathematical result about modular arithmetic . It describes the solutions to a system of linear congruences with distinct moduli . As well as being a fundamental tool in number theory , the Chinese remainder theorem forms the theoretical basis of algorithms for storing integers and in cryptography . The Chinese remainder theorem can be generalized to a statement about commutative rings ; for more about this, see the "Advanced" subpage. Contents Theorem statement Methods of proof Existence proof Explicit construction Theorem statement The Chinese remainder theorem: Let be pairwise relatively prime positive integers, and set

9. Wapedia - Wiki: Chinese Remainder Theorem Sep 25, 2010 The Chinese remainder theorem is a result about congruences in number Special cases of the Chinese remainder theorem were also known to http://wapedia.mobi/en/Chinese_remainder_theorem

Wiki: Chinese remainder theorem The Chinese remainder theorem is a result about congruences in number theory and its generalizations in abstract algebra Contents: 1. Theorem statement 2. A constructive algorithm to find the solution 3. Statement for principal ideal domains 4. Statement for general rings ... 9. External links 1. Theorem statement The original form of the theorem, contained in a third-century AD book Sun Zi suanjing (孫子算經 The Mathematical Classic by Sun Zi) by Chinese mathematician Sun Tzu and later republished in a 1247 book by Qin Jiushao , the Shushu Jiuzhang Mathematical Treatise in Nine Sections ) is a statement about simultaneous congruences (see modular arithmetic Suppose n n n k are positive integers which are pairwise coprime . Then, for any given set of integers a a a k , there exists an integer x solving the system of simultaneous congruences Furthermore, all solutions x to this system are congruent modulo the product N n n n k Hence for all , if and only if Sometimes, the simultaneous congruences can be solved even if the n i s are not pairwise coprime. A solution

10. Chinese Remainder Theorem - Exampleproblems Simultaneous congruences of integers . The original form of the theorem, contained in a thirdcentury book by Chinese mathematician Sun Tzu and later republished in a 1247 book by Qin http://www.exampleproblems.com/wiki/index.php/Chinese_remainder_theorem

Chinese remainder theorem From Exampleproblems Jump to: navigation search The Chinese remainder theorem CRT ) is the name for several related results in abstract algebra and number theory Contents Simultaneous congruences of integers Statement for principal ideal domains Statement for general rings Applications of the Chinese remainder theorem ... External links Simultaneous congruences of integers The original form of the theorem, contained in a third-century book by Chinese mathematician Sun Tzu and later republished in a book by Qin Jiushao , is a statement about simultaneous congruences (see modular arithmetic ). Suppose n n k are positive integers which are pairwise coprime (meaning gcd n i n j ) = 1 whenever i j ). Then, for any given integers a a k , there exists an integer x solving the system of simultaneous congruences Furthermore, all solutions x to this system are congruent modulo the product n n n k A solution x can be found as follows. For each i ; the integers n i and n n i are coprime, and using the extended Euclidean algorithm we can find integers r and s such that r n i s n n i = 1. If we set

11. Chinese Remainder Theorem - Discussion And Encyclopedia Article. Who Is Chinese Chinese remainder theorem. Discussion about Chinese remainder theorem. Ecyclopedia or dictionary article about Chinese remainder theorem. http://www.knowledgerush.com/kr/encyclopedia/Chinese_remainder_theorem/

12. Chinese Remainder Theorem Aug 16, 2008 The Chinese Remainder Theorem, among other things, simply provides us with one more tool for our toolbox. One more way of representing http://www.dragonwins.com/crypto/chinese_remainder_theorem.htm

DragonWins Home Page Crypto Home Chinese Remainder Theorem (Last Mod: 16 August 2008 19:11:35 One More Means of Representation Introduction by Way of Example A Bit (Hopefully Quite a Bit) More Insight Development of the Chinese Remainder Theorem ... CRT Examples One More Means of Representation We can represent numbers in a variety of different ways. For instance, the number 1925 can be written in any of the following formats: One thousand nine hundred twenty five MCMXXV Each of these representations has strengths and weaknesses - there is no "best" way to represent a number. Indeed, if that were the case we would have long ago settled on that one representation and any others would now be little more than archeological curiosities (and there are countless representations that have become just that). That so many different representations exist is a testament to the fact that most of them still serve useful purposes. How useful? Useful enough to justify the inevitable confusion that frequently results by having so many. So which representation should we use? That depends on what we are trying to do at the moment. The more representations we know about, know how to work with, and understand the strengths and weaknesses of, the more options available to us to accomplish our goal more efficiently - or perhaps even at all.

13. Chinese Remainder Theorem - Simple English Wikipedia, The Free Encyclopedia The Chinese remainder theorem is a theorem from number theory. It is about congruence. The original form was. The Chinese remainder theorem is used in cryptography, for example for http://simple.wikipedia.org/wiki/Chinese_remainder_theorem

Chinese remainder theorem From Wikipedia, the free encyclopedia Jump to: navigation search The Chinese remainder theorem is a theorem from number theory . It is about congruence . The original form was How many soldiers are there in Han Xing's army? - If you let them parade in rows of 3 soldiers, two soldiers will be left. If you let them parade in rows of 5, 3 will be left, and in rows of 7, 2 will be left? The Chinese remainder theorem is used in cryptography , for example for the RSA algorithm change Other websites Chinese remainder theorem at cut-the-knot "Chinese Remainder Theorem" by Ed Pegg, Jr., The Wolfram Demonstrations Project, 2007 This short article about mathematics or a similar topic can be made longer. You can help Wikipedia by adding to it Retrieved from " http://simple.wikipedia.org/wiki/Chinese_remainder_theorem Category Number theory Hidden category: Math stubs Personal tools New features Log in / create account Namespaces Page Talk Variants Views Read Change View history Actions Search Getting around Main Page Simple start Simple talk New changes ... Give to Wikipedia Print/export Make a book Download as PDF Page for printing Toolbox What links here Related changes Special pages Permanent link ... Cite this page In other languages Català Česky Deutsch English ... Tiếng Việt This page was last changed on 20 August 2010, at 15:28.

14. Talk:Chinese Remainder Theorem - Wikipedia, The Free Encyclopedia Please update this rating as the article progresses, or if the rating is inaccurate. Click to show/hide comments. Please add to or update the comments to suggest improvements http://en.wikipedia.org/?title=Talk:Chinese_remainder_theorem

15. Chinese Remainder Theorem - ENotes.com Reference Get Expert Help. Do you have a question about the subject matter of this article? Hundreds of eNotes editors are standing by to help. http://www.enotes.com/topic/Chinese_remainder_theorem

16. Chinese Remainder Theorem By The SwissPedia, The Free Encyclopedia By Just Click Chinese remainder theorem. The biggest multilingual freecontent encyclopedia on the Internet. Over 7 million articles in over 200 languages, and still growing. http://www.swisscorner.com/wiki.php?title=Chinese_remainder_theorem

17. Chinese Remainder Theorem Chinese remainder theorem The Chinese remainder theorem is the name applied to a number of related results in abstract algebra and number theory. http://www.fact-index.com/c/ch/chinese_remainder_theorem.html

Main Page See live article Alphabetical index Chinese remainder theorem The Chinese remainder theorem is the name applied to a number of related results in abstract algebra and number theory Table of contents 1 Simultaneous congruences of integers 2 Statement for principal ideal domains 3 Statement for general rings Simultaneous congruences of integers The original form of the theorem, contained in a book by the Chinese mathematician Ch'in Chiu-Shao published in , is a statement about simultaneous congruences (see modular arithmetic ). Suppose n n k are positive integers which are pairwise coprime (meaning gcd n i n j ) = 1 whenever i j ). Then, for any given integers a a k , there exists an integer x solving the system of simultaneous congruences x a i mod n i ) for i k Furthermore, all solutions x to this system are congruent modulo the product n n n k A solution x can be found as follows. For each i , the integers n i and n n i are coprime, and using the extended Euclidean algorithm we can find integers r and s such that r n i s n n i = 1. If we set e i s n n i , then we have e i mod n i ) and e i mod n j ) for j i The number x i k a i e i then solves the given system of simultaneous congruences.

18. Chinese Remainder Theorem - China-related Topics CE-CH - China-Related Topics The Chinese remainder theorem is any of a number of related results in abstract algebra and number theory. Simultaneous congruences of integers The original form of the http://www.famouschinese.com/virtual/Chinese_remainder_theorem

October 31, 2010 1 Introduction Simultaneous congruences of integers Statement for principal ideal domains Statement for general rings External links ... China-related Topics CE-CH Chinese remainder theorem Wikipedia The Chinese remainder theorem is any of a number of related results in abstract algebra and number theory. [go back to top] Simultaneous congruences of integers The original form of the theorem, contained in a book by the Chinese mathematician Qin Jiushao published in 1247, is a statement about simultaneous congruences (see modular arithmetic). Suppose n n k n i n j i j ). Then, for any given integers a a k x solving the system of simultaneous congruences Furthermore, all solutions x to this system are congruent modulo the product n n n k A solution x can be found as follows. For each i ; the integers and n are coprime, and using the extended Euclidean algorithm we can find integers r and s such that s n = 1. If we set s n , then we have for j i The solution to the system of simultaneous congruences is therefore For example, consider the problem of finding an integer

19. Chinese Remainder Theorem From Interactive Mathematics Miscellany And Puzzles Chinese Remainder Theorem. Application of Modular Arithmetic. According to D. Wells, the following problem was posed by Sun Tsu SuanChing (4th century AD) http://www.cut-the-knot.org/blue/chinese.shtml

20. Solving Congruences: The Chinese Remainder Theorem Sep 11, 2000 This is done by the Chinese Remainder Theorem, socalled because it Chinese Remainder Theorem For relatively prime moduli m and n, http://www.math.okstate.edu/~wrightd/crypt/lecnotes/node21.html

Next: Challenges! Up: Cryptology Class Notes Previous: Square roots Solving Congruences: The Chinese Remainder Theorem In considering the problem of finding modular square roots, we found that the problem for a general modulus m could be reduced to that for a prime power modulus. The next problem would be how to piece the solutions for prime powers together to solve the original congruence. This is done by the Chinese Remainder Theorem, so-called because it appeared in ancient Chinese manuscripts. A typical problem is to find integers x that simultaneously solve It's important in our applications that the two moduli be relatively prime; otherwise, we would have to check that the two congruences are consistent. The Chinese Remainder Theorem has a very simple answer: Chinese Remainder Theorem: For relatively prime moduli m and n , the congruences have a unique solution x modulo mn Our example problem would have a unique solution modulo It's better than this; there is a relatively simple algorithm to find the solution. Since all number theory algorithms ultimately come down to Euclid's algorithm, you can be sure it happens here as well. First let's consider an even simpler example. Suppose we want all numbers

Page 1     1-20 of 90    1  | 2  | 3  | 4  | 5  | Next 20