21. Chinese Remainder Theorem -- From Wolfram MathWorld Let r and s be positive integers which are relatively prime and let a and b be any two integers. Then there is an integer N such that N=a (mod r) (1) and N=b (mod s). (2 http://mathworld.wolfram.com/ChineseRemainderTheorem.html

Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Interactive Demonstrations Chinese Remainder Theorem Let and be positive integers which are relatively prime and let and be any two integers . Then there is an integer such that and Moreover, is uniquely determined modulo . An equivalent statement is that if , then every pair of residue classes modulo and corresponds to a simple residue class modulo The Chinese remainder theorem is implemented in Mathematica as ChineseRemainder a a m m ]. The Chinese remainder theorem is also implemented indirectly using Reduce in with a domain specification of Integers The theorem can also be generalized as follows. Given a set of simultaneous congruences for and for which the are pairwise relatively prime , the solution of the set of congruences is where and the are determined from SEE ALSO: Congruence Congruence Equation Linear Congruence Equation REFERENCES: Flannery, S. and Flannery, D. In Code: A Mathematical Journey. London: Profile Books, pp. 123-125, 2000. Ireland, K. and Rosen, M. "The Chinese Remainder Theorem." §3.4 in

22. Chinese Remainder Theorem Chinese Remainder Theorem Chinese Remainder Theorem (CRT) The following problem was posed by Sunzi Sun Tsu (4th century AD) in the book Sunzi Suanjing There are http://www.chinapage.com/math/crt.html

Chinese Remainder Theorem Chinese Remainder Theorem (CRT) The following problem was posed by Sunzi [Sun Tsu] (4th century AD) in the book Sunzi Suanjing: There are certain things whose number is unknown. Repeatedly divided by 3, the remainder is 2; by 5 the remainder is 3; and by 7 the remainder is 2. What will be the number? The answer is hidden in the following song: Math Page

23. Math_class: Number Theory 101 (Chinese Remainder Theorem) I had forgotten, at the time, that I wanted to hit on the Chinese Remainder Theorem. So, factoring will have to wait until next class. http://www.csh.rit.edu/~pat/math/series/nt/20020926/

: Number Theory 101 (Chinese Remainder Theorem) I said, in the last lesson, that we would get into factoring during this lesson. I had forgotten, at the time, that I wanted to hit on the Chinese Remainder Theorem. So, factoring will have to wait until next class. In other news, I entirely blew creating a class for two weeks ago. And, last week, I was sick to the point of inertness for 80% of the week. My apologies for blowing the class two weeks ago. I hope the content is interesting enough to bring y'all back after this unscheduled hiatus. Greatest Common Divisor (Continued) One more interesting thing to note about the Greatest Common Divisor of two numbers (at least one of which is non-zero). The Greatest Common Divisor of two numbers is the smallest positive number which can be written as a linear combination of the two numbers. It shouldn't come as a surprise to you that the proof of this takes advantage of the Well-Ordering Principle. Just about any mathematical statement containing "smallest number" requires the Well-Ordering Principle. The proof creates a set of all positive numbers a * u + b * v where u and v are integers. It shows that the set is non-empty (one way to do this is to let

24. Chinese Remainder Theorem: Definition From Answers.com The theorem that if the integers m 1 , m 2 , …, m n are relatively prime in pairs and if b 1 , b 2 , …, b n are integers, then there exists an integer that is congruent to http://www.answers.com/topic/chinese-remainder-theorem

25. The Prime Glossary: Chinese Remainder Theorem Welcome to the Prime Glossary a collection of definitions, information and facts all related to prime numbers. This pages contains the entry titled 'Chinese remainder http://primes.utm.edu/glossary/xpage/ChineseRemainderTheorem.html

Chinese remainder theorem (another Prime Pages ' Glossary entries) Glossary: Index Search Home Previous ... Mail Editor Prime Pages: Home Search Index FAQ Top 5000: Primes Provers Curios The following theorem is traditionally known as the Chinese remainder theorem (though there is some evidence that it was known to the Greeks before the Chinese). Theorem. Let n n n k are pairwise relatively prime integers. If a a a k are any integers, then There exists an integer a such a a i mod n i ) for each i k , and If b a i (mod n i ) for each i k , then b a (mod n n n k It is said that the ancient Chinese used a variant of this theorem to count their soldiers by having them line up in rectangles of 7 by 7, 11 by 11, ... After counting only the remainders, they solved the associated system of equations for the smallest positive solution. Chris K. Caldwell

26. Generalized Chinese Remainder Theorem For Groups Author Message; Jamie. Registered User. Joined 02 Aug 03. Posts 5. Location New York, US. Posted Sat Aug 02, 2003 814 pm ; Post subject generalized chinese remainder http://www.algebra-answer.com/algebra-helper/generalized-chinese-remainder-theor

@import url("images/styles.css"); @import url("forum.css"); GENERALIZED CHINESE REMAINDER THEOREM FOR GROUPS gebra homework help gebra textbooks ged algebra practice tests general algebra rules ... gebra Author Message Jamie Registered User Joined: 02 Aug 03 Posts: 5 Location: New York, US Posted: Sat Aug 02, 2003 8:14 pm ; Post subject: generalized chinese remainder theorem for groups Help me out here dude! I've got all this homework to do and I'm stuck on generalized chinese remainder theorem for groups . Where can I get help? Back to top Profile PM WWW Author Message moderator Joined: 11 Jan 2003 Posts: 1264 Location: Salt Lake City, UT Posted: Sat Aug 02, 2003 8:43 pm ; Post subject: RE: generalized chinese remainder theorem for groups The best way to figure this stuff out is to sit down on your own time and learn at your own pace. Fortunately, the Algebra Helper software lets you do just that. You can enter in your own algebra problems, and it works with you to solve them faster and make them easier to understand. It seems like algebra homework can take forever. It seems like all you do is sit there and stare at the equations hoping that it will just suddenly sink in. And yet…when you're done staring at the homework, you still probably have a ton of other homework to do. Many algebra tutorials teach you how to solve math equations that barely look like what you're trying to do. When you're done solving those, you're still left wondering how to solve yours.

27. PlanetMath: Chinese Remainder Theorem Nov 16, 2001 The Chinese remainder theorem originated in the book ``Sun Zi Suan Jing , or Sun Tzu s Arithmetic Classic, by the Chinese mathematician Sun http://planetmath.org/encyclopedia/ChineseRemainderTheorem.html

(more info) Math for the people, by the people. donor list find out how Encyclopedia Requests ... Advanced search Login create new user name: pass: forget your password? Main Menu sections Encyclopædia Papers Books Expositions meta Requests Orphanage Unclass'd Unproven ... Classification talkback Polls Forums Feedback Bug Reports downloads Snapshots PM Book information News Docs Wiki ChangeLog ... About Chinese remainder theorem (Theorem) Suppose we have a set of $n$ congruences of the form where are relatively prime . Let and, for all ), let be an integer that satisfies Then one solution of these congruences is Any satisfies the set of congruences if and only if it satisfies 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 owner history view preamble ... get metadata View style: jsMath HTML HTML with images page images TeX source 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 Log in to rate this entry.

28. Chinese Remainder Theorem Chinese remainder theorem refers to a result about congruences in number theory and its generalizations in abstract algebra. Theorem statement The original form of the theorem http://english.turkcebilgi.com/Chinese remainder theorem

EnglishInfo Search Türkçe Home Information Translation ... Chinese remainder theorem Information about Chinese remainder theorem Chinese remainder theorem Information about Chinese remainder theorem Double click any English word, to find Turkish meaning Chinese remainder theorem refers to a result about congruences in number theory and its generalizations in abstract algebra Theorem statement The original form of the theorem, contained in a third-century AD 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 n k are integers which are pairwise coprime . Then, for any given 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 Sometimes, the simultaneous congruences can be solved even if the n i 's are not pairwise coprime. A solution x exists if and only if: All solutions x are then congruent modulo the least common multiple of the n i Versions of the Chinese remainder theorem were also known to Brahmagupta , and appear in Fibonacci 's Liber Abaci A constructive algorithm to find the solution This algorithm only treats the situations where the 's are coprime. The

29. The Chinese Remainder Theorem The Chinese Remainder Theorem says that certain systems of simultaneous congruences with different moduli have solutions. The idea embodied in the theorem http://marauder.millersville.edu/~bikenaga/numbertheory/chinese-remainder/chines

The Chinese Remainder Theorem The Chinese Remainder Theorem gives solutions to systems of congruences with relatively prime moduli. The solution to a system of congruences with relatively prime moduli may be produced using a formula by computing modular inverses, or using an iterative procedure involving successive substitution. The Chinese Remainder Theorem says that certain systems of simultaneous congruences with different moduli have solutions. The idea embodied in the theorem was apparently known to Chinese mathematicians a long time ago - hence the name. I'll begin by collecting some useful lemmas. Lemma 1. Let m and be positive integers. If m is relatively prime to each of , then it is relatively prime to their product Proof. If , then there is a prime p which divides both m and . Since , p must divide for some i. Now p divides both m and , so Example. 6 is relatively prime to 25, to 7, and to 11. , and I showed earlier that the greatest common divisor of a and b is greatest in the sense that it is divisible by any common divisor of a and b. The next result is the analogous statement for least common multiples.

30. Chinese Remainder Theorem Chinese remainder theorem Linear Abstract Algebra discussion Vote for Physics Forums in the physics.org Best Q A site category! (you have to register) Vote here http://www.physicsforums.com/showthread.php?t=277652

31. MATLAB Central - File Detail - Chinese Remainder Theorem For Integers Chinese Remainder Theorem for Integers. by Sundar Krishnan . 09 Sep 2004 (Updated 14 Sep 2004) No BSD License http://www.mathworks.com/matlabcentral/fileexchange/5851-chinese-remainder-theor

Search: MATLAB Central File Exchange MATLAB Newsgroup Link Exchange Blogs Programming Contest MathWorks.com All of MathWorks.com Login Create Account Login File Exchange ... Rate this file 10 downloads (last 30 days) File Size: 32.74 KB File ID: #5851 Chinese Remainder Theorem for Integers by Sundar Krishnan 09 Sep 2004 (Updated 14 Sep 2004) No BSD License This programme verifies the Chinese Remainder Theorem for Integers (of "congruence"). Download Now Watch this File File Information Description This programme verifies the Chinese Remainder Theorem for Integers (of "congruence"). Assume that we are to find a solution c_soln such that : c_soln =eqvt mod (4, 7) c_soln =eqvt mod (5, 17) c_soln =eqvt mod (6, 23) c_soln =eqvt mod (7, 47) where "=eqvt" implies "congruence" with the usual symbol of 3 equal-to lines. The solution is c_soln = 124275 It can be verified by for eg : mod (124275, 7) = 4 ; mod (124275, 17) = 5 ; Now, how did we find this c_soln ?

32. The Chinese Remainder Theorem With Limits « Gödel’s Lost Letter And P=NP Coppersmith’s theorem and adding size constraints to the Chinese Remainder Theorem . Sun Tzu is famous for the discovery of the Chinese Remainder Theorem (CRT) in China in http://rjlipton.wordpress.com/2009/08/01/the-chinese-remainder-theorem-with-limi

Skip to content a personal view of the theory of computation Home About Me About P=NP and SAT Conventional Wisdom and P=NP ... Thank You Page The Chinese Remainder Theorem With Limits August 1, 2009 tags: Algorithms BDD chinese remainder theorem CRT ... time by rjlipton Sun Tzu is famous for the discovery of the Chinese Remainder Theorem (CRT) in China in the third century, way before it was known in the west. His original example was: , for example, is a direct product , but that is not very helpful. For example, suppose we need to understand the structure of the solution set to equations like where are boolean and is a polynomial. The CRT does not seem to help here. The problem is this: the direct product of two boolean vectors need not be boolean. For instance, a vector can be boolean modulo and boolean modulo , but not boolean modulo I recently posted on a paper that will appear at FOCS on a related problem. Solving an Equation Modulo There is a wonderful theorem, due to Coppersmith, that should be in all our tool boxes. Theorem: Let be a natural number and a monic polynomial of degree . Set for some . Then, there is an algorithm to find all

33. Chinese Remainder Theorem File Format PDF/Adobe Acrobat Quick View http://www.cargalmathbooks.com/12 Chinese Remainder Theorem .pdf

34. Nrich.maths.org :: Mathematics Enrichment :: The Chinese Remainder Theorem Now let s move on to the Chinese Remainder Theorem itself. Theorem. Let $p_1$, $ p_2$, , and $p_n$ be distinct primes. For any integers $a_1$, $a_2$, http://nrich.maths.org/5466

Skip over navigation Search by Topic Tech Help Thesaurus ... Contact Us Be part of our research! Are you really good at maths? Are you the parent of someone who is really good at maths? Are you the teacher of someone who is really good at maths? If so we'd love to involve you in our research. Case studies Are you aged between 10 and 17? Read about Dr Yi Feng's research which involves talking to students over the next two years about their mathematical experiences in and out of school, and respond to the call for participants. AskNRICH Do you use the ASKNRICH site, or have you previously? Libby Jared's research involves looking at what goes on on the ASKNRICH site and why. To help with this research please respond to the ASKNRICH survey Read more about the research we are doing here This month: The Chinese Remainder Theorem Article Printable page Article by Vicky Neale Stage: 4 and 5 In this article we shall consider how to solve problems such as Find all integers that leave a remainder of $1$ when divided by $2$, $3$, and $5$. You might like to think about this problem for yourself before reading on.

35. CHINESE REMAINDER THEOREM The ChineseRemainder Theorem involvesa situation like the following we are asked to http://www.math.hawaii.edu/~lee/courses/Chinese.pdf

36. Chinese Remainder Theorem - Wolfram Demonstrations Project For a parade, marchers are arranged in columns of seven, but one person is left out. In columns of eight, two people are left out. With columns of nine, three people are left out. http://demonstrations.wolfram.com/ChineseRemainderTheorem/

37. Chinese Remainder Theorem - What Does CRT Stand For? Acronyms And Abbreviations Acronym Definition; CRT Cathode Ray Tube CRT Credit CRT Court (Canada Post road designation) CRT Civil Rights (Division, US Department of Health Human Services) http://acronyms.thefreedictionary.com/Chinese remainder theorem

38. CHINESE REMAINDER THEOREM by C Ding (Turku Centre for Computer Science, Finland), D Pei (Chinese Academy of Science, Beijing), A Salomaa (University of Turku, Finland) http://www.worldscibooks.com/compsci/3254.html

Search Home Join Our Mailing List New Reviews New Titles ... E-Catalogues COMPUTER SCIENCE Artificial Intelligence Database/ Information Sciences Decision Sciences Digital Security Fuzzy Logic Machine Vision/ Pattern Recognition Neural Networks/ Networking Parallel Processing/ Supercomputing Software Engineering Theoretical Computer Science General New Titles ... Book Series Related Journals International Journal of Semantic Computing (IJSC) International Journal of Information Acquisition (IJIA) Computer Science Journals New Mathematics and Natural Computation (NMNC) ... Request for related catalogues PRODUCTS Journals eBooks Journals Archives eProceedings RESOURCES Print flyer Full Version Condensed Version Recommend title For Librarians ... News CHINESE REMAINDER THEOREM Applications in Computing, Coding, Cryptography by C Ding (Turku Centre for Computer Science, Finland) D Pei (Chinese Academy of Science, Beijing) A Salomaa (University of Turku, Finland) Table of Contents Preface Chapter 1: Introduction and Philosophy Chinese Remainder Theorem , CRT, is one of the jewels of mathematics. It is a perfect combination of beauty and utility or, in the words of Horace

39. Chapter 7: The Chinese Remainder Theorem The Chinese Remainder Theorem. 7.1 Solving Two Congruences 7.2 A More General Theorem 7.3 Solving Lots of Congruences 7.4 Explicit Formulas http://www.math.mtu.edu/mathlab/COURSES/holt/dnt/chinese.html

Chapter 7 The Chinese Remainder Theorem Solving Two Congruences A More General Theorem Solving Lots of Congruences Explicit Formulas ... DNT Table of Contents

40. Chinese Remainder Theorem (mathematics) -- Britannica Online Encyclopedia Chinese remainder theorem (mathematics), ancient theorem that gives the conditions necessary for multiple equations to have a simultaneous integer solution. The theorem has its http://www.britannica.com/EBchecked/topic/112749/Chinese-remainder-theorem

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