Chinese Remainder Theorem -- From MathWorld chinese remainder theorem, In a future version of Mathematica, the Chinese remaindertheorem will be implemented indirectly using the Reduce command. http://mathworld.wolfram.com/ChineseRemainderTheorem.html
Extractions: Moreover, N is uniquely determined modulo rs . An equivalent statement is that if , then every pair of residue classes modulo r and s corresponds to a simple residue class modulo rs The Chinese remainder theorem is implemented as ChineseRemainder a a m m Mathematica add-on package NumberTheory`NumberTheoryFunctions` (which can be loaded with the command ). In a future version of Mathematica , the Chinese remainder theorem will be implemented indirectly using the Reduce command. The theorem can also be generalized as follows. Given a set of simultaneous congruences
CHINESE REMAINDER THEOREM chinese remainder theorem. Applications in Computing, Coding, Cryptography http://www.wspc.com/books/compsci/3254.html
Extractions: 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.
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.nist.gov/dads/HTML/chineseRmndr.html
Extractions: (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 Jul 16 10:09:06 1999.
Chinese Remainder Theorem chinese remainder theorem. Problems of this kind are all examples ofwhat universally became known as the chinese remainder theorem. http://www.cut-the-knot.com/blue/chinese.shtml
Extractions: Recommend this site 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
Chinese Remainder Theorem chinese remainder theorem. Application of Modular Arithmetic. of what universally became known as the chinese remainder theorem. In mathematical parlance the problems can be http://www.cut-the-knot.com/blue/chinese.html
Extractions: Recommend this site 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
CTK Exchange Subject Re chinese remainder theorem Date Tue, 2 Sep 1997 0000590400 From Alex Bogomolny. Dear Tan Yours is an example of http://www.cut-the-knot.com/exchange/chinese2.shtml
Extractions: Dear Tan: Yours is an example of problems solved in general case by what's known as the Chinese Remainder Theorem. You can look it up in O.Ore, "Number Theory and Its History", or H.Davenport, "The Higher Arithmetic" Both available through my bookstore. In your particular case, you are looking for a number X such that X = 1 (mod 2,3,4) and X = (mod 5) which means that divided by 2,3,4 X has the remainder 1 while divided by 5 the remainder is 0. The first three condition say that (X - 1) is divided by 2,3 and 4, i.e., by their least common multiple which is 12. Therefore, X - 1 = 12t for some integer t. From X = (mod 5) it follows that X - 1 = 4 (mod 5). Or 12t = 4(mod 5), 3t = 1 (mod 5). As you can check then, t = 5k + 2 for an integer k. Combining this with X = 12t + 1 we get X = 60k + 25. There are three numbers below 200 in this form: 25, 85 and 145. Best regards
Chinese Remainder Theorem chinese remainder theorem http://linguistlist.org/~zheng/courseware/remainder.html
Chinese Remainder Theorem -- From MathWorld chinese remainder theorem, In a future version of Mathematica, the Chinese remaindertheorem will be implemented indirectly using the Reduce command. http://mathworld.wolfram.com/C/ChineseRemainderTheorem.html
Extractions: Moreover, N is uniquely determined modulo rs . An equivalent statement is that if , then every pair of residue classes modulo r and s corresponds to a simple residue class modulo rs The Chinese remainder theorem is implemented as ChineseRemainder a a m m Mathematica add-on package NumberTheory`NumberTheoryFunctions` (which can be loaded with the command ). In a future version of Mathematica , the Chinese remainder theorem will be implemented indirectly using the Reduce command. The theorem can also be generalized as follows. Given a set of simultaneous congruences
The Chinese Remainder Theorem The chinese remainder theorem. Last updated August 7th, 1995 Thechinese remainder theorem (CRT) gives the answer to the problem http://www.jjj.de/mtommila/crt.html
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 theorem.' Come explore a new prime term today! chinese remainder theorem. (another Prime Pages' Glossary entries) http://www.utm.edu/research/primes/glossary/ChineseRemainderTheorem.html
Extractions: (another Prime Pages ' Glossary entries) Glossary: Prime Pages: 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). 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.
The Chinese Remainder Theorem The chinese remainder theorem. We prove the chinese remainder theorem andThue's Theorem as well as several useful number theory propositions. http://mizar.uwb.edu.pl/JFM/Vol9/wsierp_1.html
Extractions: Association of Mizar Users The terminology and notation used in this paper have been introduced in the following articles [ Contents (PDF format) 1] Grzegorz Bancerek. The fundamental properties of natural numbers Journal of Formalized Mathematics 2] Grzegorz Bancerek. Joining of decorated trees Journal of Formalized Mathematics 3] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences Journal of Formalized Mathematics 4] Czeslaw Bylinski. Functions and their basic properties Journal of Formalized Mathematics 5] Czeslaw Bylinski. The sum and product of finite sequences of real numbers Journal of Formalized Mathematics 6] Library Committee. Introduction to arithmetic Journal of Formalized Mathematics Addenda 7] Krzysztof Hryniewiecki. Basic properties of real numbers Journal of Formalized Mathematics 8] Katarzyna Jankowska. Transpose matrices and groups of permutations Journal of Formalized Mathematics 9] Andrzej Kondracki.
Extractions: 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: 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
Chinese Remainder Theorem - Wikipedia chinese remainder theorem. The chinese remainder theorem is the name appliedto a number of related results in abstract algebra and number theory. http://www.wikipedia.org/wiki/Chinese_remainder_theorem
Chinese Remainder Theorem - Wikipedia chinese remainder theorem. (Redirected from chinese remainder theorem).The chinese remainder theorem is the name applied to a number http://www.wikipedia.org/wiki/Chinese_Remainder_Theorem
Chinese Remainder Theorem next up previous contents Next Exercises Up Congruences Previous Exercises.chinese remainder theorem. Proof. We first construct a solution. http://www.math.swt.edu/~haz/prob_sets/notes/node25.html
Extractions: Next: Exercises Up: Congruences Previous: Exercises Proof. We first construct a solution. Let and, for each i . Note that for every i . Thus, has a solution . Define Since we see that To see the uniqueness, Let x ' be another solution. Then for each i . Noting that all 's are pairwise relatively prime, we have that , i.e., the solution x is unique.
About "Chinese Remainder Theorem" chinese remainder theorem. Library Home Full Table of Contents Suggest a Link Library Help Visit this site http//www.cut http://mathforum.org/library/view/6932.html
Extractions: Visit this site: http://www.cut-the-knot.com/blue/chinese.html Author: Interactive Mathematics Miscellany and Puzzles, Alexander Bogomolny Description: An explanation and proof, using modular arithmetic, of the Chinese Remainder Theorem, which concerns problems of the following type: 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? Levels: High School (9-12) College Languages: English Resource Types: Articles Math Topics: Basic Algebra Basic Operations History and Biography Number Theory ... Search
Chinese Remainder Theorem chinese remainder theorem. This is an engin to solve a kind of ChineseRemainder problem by using the method described in Page 137 http://www.linguistlist.org/~zheng/courseware/remainder.html
Chinese Remainder Theorem chinese remainder theorem. Author hasinoff What is the chinese remainder theoremas it applies to solving equations involving the modulus operator? http://newton.dep.anl.gov/newton/askasci/1995/math/MATH056.HTM
Chinese Remainder Theorem chinese remainder theorem. Find the two smallest counting numbers that will eachhave the remainders 2, 3, and 2 when divided by 3, 5, and 7 respectively. http://pegasus.cc.ucf.edu/~ucfcasio/pow/chinese.htm
Extractions: Yona Levine. Lehava, Kedumim, Israel Xingji Zheng. Abby Senior SS, Abbotsford, BC, Canada Michael Moyer. The Way Home School, Carlisle, PA Bella Voldman. Brookline HS, Brookline, MA Stephan Wild. BSZ 3, Leipzig, Germany Shu Duan. Ecole Marie-Esther, Shippagan, New Brunswick, Canada Lisa Goliber. Spalding Catholic, Granville, IA Edgar Pantoja. Carteret HS, Carteret, NJ Wojciech Lewkowicz. Lemont HS, Lemont, IL Amanda Vicary. Farmington HS, Farmington, IL Paul Pollack. Gulf HS, New Port Richey, FL Gregory Winston. O'Neill CVI, Oshawa, Ontario, Canada David Sorani. Shaare HS, Brooklyn, NY Ido Yariv. Gan-Nachum School, Rishon LeZion, Israel Kenny Ho. Gordon Graydon SS, Mississauga, Ontario, Canada Katie Dawson. Newnan HS, Newnan, GA Amit Sahasrabudhe. TL Kennedy, Mississauga, Ontario, Canada Sameer Akhtar. TL Kennedy, Mississauga, Ontario, Canada Fang Yi Liu. South Hills HS, Covina, CA
