1. Chinese Remainder Theorem chinese remainder theorem. Application of Modular Arithmetic. http://www.cut-the-knot.org/blue/chinese.shtml

Username: Password: Sites for teachers Sites for parents Awards Interactive Activities ... Sites for parents Chinese Remainder Theorem According to D.Wells, the following problem was posed by Sun Tsu Suan-Ching (4th century AD): 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? Oystein Ore mentions another puzzle with a dramatic element from Brahma-Sphuta-Siddhanta (Brahma's Correct System) by Brahmagupta (born 598 AD): An old woman goes to market and a horse steps on her basket and crashes the eggs. The rider offers to pay for the damages and asks her how many eggs she had brought. She does not remember the exact number, but when she had taken them out two at a time, there was one egg left. The same happened when she picked them out three, four, five, and six at a time, but when she took them seven at a time they came out even. What is the smallest number of eggs she could have had? Problems of this kind are all examples of what universally became known as the Chinese Remainder Theorem . In mathematical parlance the problems can be stated as finding n, given its remainders of division by several numbers m

2. The Prime Glossary Chinese Remainder Theorem chinese remainder theorem (another Prime Pages' Glossary entries) Glossary Index. Search. Home. Previous. Next. Random. Mail Editor Prime http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

3. Chinese Remainder Theorem -- From MathWorld The chinese remainder theorem is implemented as ChineseRemainder{a1, a2, }{m1,m2, } in the Mathematica addon package http://mathworld.wolfram.com/ChineseRemainderTheorem.html

INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index DESTINATIONS About MathWorld About the Author Headline News ... Random Entry CONTACT Contribute an Entry Send a Message to the Team MATHWORLD - IN PRINT Order book from Amazon Number Theory Congruences 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 as ChineseRemainder a a m m ] in the Mathematica add-on package NumberTheory`NumberTheoryFunctions` (which can be loaded with the command ) . The Chinese remainder theorem is also implemented indirectly using Reduce starting in Mathematica Version 5.0 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 [Pages Linking Here] REFERENCES: Flannery, S. and Flannery, D.

4. Math 5410 Chinese Remainder Theorem chinese remainder theorem Theorem Suppose that m 1 , m 2 , , m r are pairwise relatively prime positive integers, and let a 1 http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

5. Chinese Remainder Theorem chinese remainder theorem "We have a number of things, but we do not know exactly how many. If we count them by threes we have two left over. http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

6. The Prime Glossary: Chinese Remainder Theorem Welcome to the Prime Glossary a collection of definitions, information and factsall related to prime numbers. This pages contains the entry titled http://primes.utm.edu/glossary/page.php?sort=ChineseRemainderTheorem

7. Chinese Remainder Theorem chinese remainder theorem. Author hasinoff What is the chinese remainder theorem as it applies to solving equations involving the modulus operator? http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

8. The Chinese Remainder Theorem The chinese remainder theorem. Last updated August 7th, 1995. The ChineseRemainder Theorem (CRT) gives the answer to the problem http://www.apfloat.org/crt.html

The Chinese Remainder Theorem Last updated: August 7th, 1995 The Chinese Remainder Theorem (CRT) gives the answer to the problem: Find the number x, that satisfies all the n equations simultaneously: x = r1 (mod p1) x = r2 (mod p2) x = rk (mod pk) x = rn (mod pn) We will assume here (for practical purposes) that the moduli pk are primes. Then there exists a unique solution x modulo p1*p2*...*pn. The solution can be found with the following algorithm: Let P=p1*p2*...*pn Let the numbers T1...Tn be defined so that for each Tk (k=1...n) (P/pk)*Tk=1 (mod pk) that is, Tk is the inverse of P/pk (mod pk). The inverse of a (mod p) can be found for example by calculating a^(p-2) (mod p). Note that a*a^(p-2)=a^(p-1)=1 (mod p). Then the solution is x = (P/p1)*r1*T1 + (P/p2)*r2*T2 + ... + (P/pn)*rn*Tn (mod P) The good thing is, that you can calculate the factors (P/pk)*Tk beforehand, and then to get x for different rk, you only need to do simple multiplications and additions (supposing that the primes pk remain the same). When using the CRT in a number theoretic transform, the algorithm can be implemented very efficiently using only single-precision arithmetic when rk

9. The Chinese Remainder Theorem The chinese remainder theorem http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

10. Math 5410 Chinese Remainder Theorem chinese remainder theorem. Theorem Suppose that m1, m2, , mr are pairwiserelatively prime positive integers, and let a1, a2, , ar be integers. http://www-math.cudenver.edu/~wcherowi/courses/m5410/ctccrt.html

Chinese Remainder Theorem Theorem : Suppose that m , m , ..., m r are pairwise relatively prime positive integers, and let a , a , ..., a r be integers. Then the system of congruences, x = a i (mod m i x m x ... x m r , which is given by: x = a M y + a M y + ... + a r M r y r (mod M), where M i = M/m i and y i = (M i (mod m i Pf : Notice that gcd(M i , m i i all exist (and can be determined easily from the extended Euclidean Algorithm). Now, notice that since M i y i = 1 (mod m i ), we have a i M i y i = a i (mod m i i M i y i = (mod m j ) if j is not i (since m j i in this case). Thus, we see that x = a i (mod m i If there were two solutions, say x , and x , then we would have x - x = (mod m i ) for all i, so x - x = (mod M), i.e., they are the same modulo M. Example Find the smallest multiple of 10 which has remainder 2 when divided by 3, and remainder 3 when divided by 7. We are looking for a number which satisfies the congruences, x = 2 mod 3, x = 3 mod 7, x = mod 2 and x = mod 5. Since, 2, 3, 5 and 7 are all relatively prime in pairs, the Chinese Remainder Theorem tells us that there is a unique solution modulo 210 ( = 2x3x5x7). We calculate the M i 's and y i 's as follows: M = 210/2 = 105; y

11. 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 http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

12. Chinese Remainder Theorem Definition of chinese remainder theorem, possibly with links to more informationand implementations. http://www.nist.gov/dads/HTML/chineseRmndr.html

Chinese remainder theorem (algorithm) Definition: An integer n can be solved uniquely mod LCM(A(i)) Note: For example, knowing the remainder of n when it's divided by 3 and the remainder when it's divided by 5 allows you to determine the remainder of n when it's divided by LCM(3,5) = 15. After LK. Author: PEB Go to the Dictionary of Algorithms and Data Structures home page. If you have suggestions, corrections, or comments, please get in touch with Paul E. Black (paul.black@nist.gov). Entry modified Fri Dec 17 12:14:16 2004. HTML page formatted Wed Aug 24 11:18:15 2005. Cite this as: Paul E. Black, "Chinese remainder theorem", from Dictionary of Algorithms and Data Structures , Paul E. Black, ed., NIST http://www.nist.gov/dads/HTML/chineseRmndr.html

13. Chinese Remainder Theorem chinese remainder theorem. Application of Modular Arithmetic. http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

14. PlanetMath: Chinese Remainder Theorem The chinese remainder theorem is said to have been used to count the size of theancient Attachments chinese remainder theorem proof (Proof) by vampyr http://planetmath.org/encyclopedia/ChineseRemainderTheorem.html

(more info) Math for the people, by the people. Encyclopedia Requests Forums Docs ... Random Login create new user name: pass: forget your password? Main Menu sections Encyclop¦dia 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 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 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 saforres full author list owner history view preamble View style: HTML with images page images TeX source Attachments: Chinese remainder theorem proof (Proof) by vampyr Cross-references: groups size solution satisfies ... congruences There are 3 references to this object.

15. Lee Lady Topics In Upper Division Mathematics The Fundamental Theorem of Arithmetic (Click here for dvi Solving the Chinese Remainder Problem (Click here for dvi format.) (Click here for http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

16. PlanetMath: Chinese Remainder Theorem chinese remainder theorem is owned by bwebste. full author list (2) owner Attachments proof of chinese remainder theorem (Proof) by mclase http://planetmath.org/encyclopedia/ChineseRemainderTheorem2.html

(more info) Math for the people, by the people. Encyclopedia Requests Forums Docs ... Random Login create new user name: pass: forget your password? Main Menu sections Encyclop¦dia 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) Let be a commutative ring with identity . If are ideals of such that whenever , then let The sum of quotient maps gives an isomorphism This has the slightly weaker consequence that given a system of congruences , there is a solution in which is unique mod , as the theorem is usually stated for the integers "Chinese remainder theorem" is owned by bwebste full author list owner history view preamble View style: HTML with images page images TeX source Attachments: proof of chinese remainder theorem (Proof) by mclase Cross-references: integers theorem solution congruences ... commutative ring There are 4 references to this object. This is version 4 of Chinese remainder theorem , born on 2002-02-03, modified 2003-09-06.

17. Chinese Remainder Problem Unfortunately, Problem 26 is the only problem that illustrates the chinese remainder theorem in the Sun Tzu Suan Ching. http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

18. Chinese Remainder Theorem - Wikipedia, The Free Encyclopedia The chinese remainder theorem is the name for several related results in abstract For a principal ideal domain R the chinese remainder theorem takes the http://en.wikipedia.org/wiki/Chinese_remainder_theorem

19. History Of Mathematics China Systematic treatment of indeterminate simultaneous linear congruences (chinese remainder theorem). Euclidean algorithm for GCD. http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

20. CHINESE REMAINDER THEOREM chinese remainder theorem Applications in Computing, Coding, Cryptography chinese remainder theorem, CRT, is one of the jewels of mathematics. http://www.worldscibooks.com/compsci/3254.html

Home Browse by Subject Bestsellers New Titles ... Browse all Subjects Search Bookshop New Titles Editor's Choice Bestsellers Book Series ... Join Our Mailing List CHINESE REMAINDER THEOREM Applications in Computing, Coding, Cryptography by C Ding (Turku Centre for Computer Science, Finland) , D Pei (Chinese Academy of Science, Beijing) (University of Turku, Finland) 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, omne tulit punctum qui miscuit utile dulci. Known already for ages, CRT continues to present itself in new contexts and open vistas for new types of applications. So far, its usefulness has been obvious within the realm of "three C's". Computing was its original field of application, and continues to be important as regards various aspects of algorithmics and modular computations. Theory of codes and cryptography are two more recent fields of application. This book tells about CRT, its background and philosophy, history, generalizations and, most importantly, its applications. The book is self-contained. This means that no factual knowledge is assumed on the part of the reader. We even provide brief tutorials on relevant subjects, algebra and information theory. However, some mathematical maturity is surely a prerequisite, as our presentation is at an advanced undergraduate or beginning graduate level. We have tried to make the exposition innovative, many of the individual results being new. We will return to this matter, as well as to the interdependence of the various parts of the book, at the end of the Introduction.

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