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Chinese Remainder Theorem:     more detail
1. OSU-CS-TR by Kim Sin Lee, 1994
2. Chinese Remainder Theorem: Applications in Computing, Coding, Cryptography by C. Ding, D. Pei, et all 1999-06

1. Chinese Remainder Theorem
chinese remainder theorem. This is an engin to solve a kind of Chinese Remainder problem by using the method described in Page 137, Elementary Number Theory and its Applications ( Third Edition 1993)
http://www.linguistlist.org/~zheng/courseware/remainder.html
##### Chinese Remainder Theorem
This is an engin to solve a kind of Chinese Remainder problem by using the method described in Page 137, Elementary Number Theory and its Applications (Third Edition,1993) by Kenneth H. Rosen. Please fill the necessary values in the form,then click OK button. x ( mod
x ( mod
x ( mod
x ( mod
x ( mod

click to get the solution. Return to CGI Coursewares for Number Theory Page.

2. Chinese Remainder Theorem
Definition of chinese remainder theorem, possibly with links to more information and implementations. chinese remainder theorem. ( algorithm) Definition An integer n can be solved uniquely mod
##### 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 Wed Dec 3 11:51:02 2003.

 3. The Prime Glossary: Chinese Remainder Theorem related to prime numbers. This pages contains the entry titled Chineseremainder theorem. Come explore a new prime term today!http://primes.utm.edu/glossary/page.php?sort=ChineseRemainderTheorem

4. Chinese Remainder Theorem
chinese remainder theorem. 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.
http://www.math.swt.edu/~haz/prob_sets/notes/node25.html
Next: Exercises Up: Congruences Previous: Exercises
##### Chinese Remainder Theorem
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.
Donald Hazlewood and Carol Hazlewood
Wed Jun 5 14:35:14 CDT 1996

5. Math 5410 Chinese Remainder Theorem
chinese remainder theorem. Theorem Suppose that m1, m2, , mr are pairwise relatively prime positive integers prime in pairs, the chinese remainder theorem tells us that there is
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

6. Math_class: Number Theory 101 (Chinese Remainder Theorem)
math_class Number Theory 101 (chinese remainder theorem) Disclaimers and Apologies. I said, in the last lesson, that we would get into factoring during this lesson. 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

7. Chinese Remainder Theorem
chinese remainder theorem. Problems of this kind are all examples ofwhat universally became known as the chinese remainder theorem.
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##### 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

8. Chinese Remainder Theorem
chinese remainder theorem. Application of Modular Arithmetic. chinese remainder theorem. According to D.Wells, the following problem was posed by Sun Tsu universally became known as the chinese remainder theorem. In mathematical parlance the
http://www.cut-the-knot.com/blue/chinese.html
CTK Exchange Front Page
Movie shortcuts

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Recommend this site
##### 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

9. 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.org/exchange/chinese2.shtml
 CTK Exchange Front Page Movie shortcuts Personal info ... Recommend this site Subject: Re: Chinese remainder theorem Date: Tue, 2 Sep 1997 00:00:59 -0400 From: Alex Bogomolny 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

10. Chinese Remainder Theorem -- From MathWorld
chinese remainder theorem.
http://mathworld.wolfram.com/ChineseRemainderTheorem.html
 INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index ABOUT THIS SITE About MathWorld About the Author DESTINATIONS What's New MathWorld Headline News Random Entry ... Live 3D Graphics CONTACT Email Comments Contribute! Sign the Guestbook MATHWORLD - IN PRINT Order book from Amazon Number Theory Congruences Chinese Remainder Theorem 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 and 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 NumberTheoryNumberTheoryFunctions (which can be loaded with the command ). The Chinese remainder theorem is also implemented indirectly using Reduce 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 i r and for which the are pairwise relatively prime , the solution of the set of congruences is where and the are determined from Congruence Congruence Equation Linear Congruence Equation search Flannery, S. and Flannery, D.

 11. Chinese Remainder Theorem From MathWorld chinese remainder theorem from 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\equiv a\ \lefthttp://rdre1.inktomi.com/click?u=http://mathworld.wolfram.com/ChineseRemainderTh

12. Chinese Remainder Problem
Unfortunately, Problem 26 is the only problem that illustrates thechinese remainder theorem in the Sun Tzu Suan Ching. As such, we
http://www.math.sfu.ca/histmath/China/3rdCenturyBC/CRP1.html
##### What is it?
These particular kinds of mathematical problem falls in the category of indeterminate analysis. Usually, it appears in the form as such (in modern notation):
N = m1x + r1 N = m2y + r2 N = m3z + r3
Or in modern number theory notation:
N r1 (mod m1) N r2 (mod m2) N r3 (mod m3)
Aside: Writing N r1 (mod m1) [this means N is congruent to r1 modulo m1] means that N divided by m1 leaves r1 as the remainder. The goal here is to find the smallest positive integer satisfying the congruences states above.
##### Origins.
Now that you know what a Chinese Remainder Problem is, you must be wondering why or what has this particular kind of problem to do with Chinese Mathematical History. The reason why it is called the Chinese Remainder Problem is because the earliest versions of these congruence problems occured in early Chinese mathematical works. The earliest of such works that contains the Chinese Remainder Problem is the Sun Tzu Suan Ching (also known as Sunzi suanjing) written in approximately late third century by Sun Zi . Problem 26 (also known as the problem of Master Sun) in the third volume of the Sun Tzu Suan Ching offers the earliest recorded Chinese Remainder Problem. Problem 26 is as stated below:
"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. If we count them by fives we have three left over. If we count them by sevens we have two left over. How many things are there?" (Quoted from Sun Tze Suan Ching).

 13. CHINESE REMAINDER THEOREM E. L. Lady chinese remainder theoremE. L. LadyThe chinese remainder theorem involves a situation like the following we are asked tohttp://www.math.hawaii.edu/~lee/courses/Chinese.pdf

14. Chinese Remainder Theorem - Wikipedia, The Free Encyclopedia
chinese remainder theorem. The chinese remainder theorem is any of anumber of related results in abstract algebra and number theory.
http://en.wikipedia.org/wiki/Chinese_remainder_theorem
##### Chinese remainder theorem
The Chinese remainder theorem is any of 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

...
edit
##### Simultaneous congruences of integers
The original form of the theorem, contained in a book by the Chinese mathematician Ch'in Chiu-Shao http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Qin_Jiushao.html 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 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

15. Chinese Remainder Theorem - Wikipedia, The Free Encyclopedia
chinese remainder theorem. (Redirected from chinese remainder theorem).The Chinese I k ) ). External links. chinese remainder theorem.
http://en.wikipedia.org/wiki/Chinese_Remainder_Theorem
##### Chinese remainder theorem
(Redirected from Chinese Remainder Theorem The Chinese remainder theorem is any of 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 4 External links ... edit
##### Simultaneous congruences of integers
The original form of the theorem, contained in a book by the Chinese mathematician Ch'in Chiu-Shao http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Qin_Jiushao.html 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 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

16. Chinese Remainder Theorem - Wikipedia, The Free Encyclopedia
http://en2.wikipedia.org/wiki/Chinese_remainder_theorem
##### Chinese remainder theorem
The Chinese remainder theorem is any of 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

...
edit
##### Simultaneous congruences of integers
The original form of the theorem, contained in a book by the Chinese mathematician Ch'in Chiu-Shao http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Qin_Jiushao.html 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 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

17. CHINESE REMAINDER THEOREM
chinese remainder theorem Applications in Computing, Coding, Cryptography by CDing (Turku Centre for Computer Science, Finland), D Pei (Chinese Academy of
http://www.worldscientific.com/books/compsci/3254.html
 Home Browse by Subject Bestsellers New Titles ... Browse all Subjects Search Keyword Author Concept ISBN Series 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.

18. 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

19. Chinese Remainder Theorem
chinese remainder theorem. The chinese remainder theorem is the name appliedto 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.

20. 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
Journal of Formalized Mathematics
Volume 9, 1997

University of Bialystok

Association of Mizar Users
##### The Chinese Remainder Theorem
Andrzej Kondracki
AMS Management Systems Poland, Warsaw
##### MML Identifier:
The terminology and notation used in this paper have been introduced in the following articles [ Contents (PDF format)
##### Bibliography
1] Grzegorz Bancerek. The fundamental properties of natural numbers Journal of Formalized Mathematics
2] Grzegorz Bancerek. The ordinal numbers Journal of Formalized Mathematics
3] Grzegorz Bancerek. Joining of decorated trees Journal of Formalized Mathematics
4] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences Journal of Formalized Mathematics
5] Czeslaw Bylinski. Functions and their basic properties Journal of Formalized Mathematics
6] Czeslaw Bylinski. The sum and product of finite sequences of real numbers Journal of Formalized Mathematics
7] Katarzyna Jankowska. Transpose matrices and groups of permutations Journal of Formalized Mathematics
8] Andrzej Kondracki. Basic properties of rational numbers Journal of Formalized Mathematics
9] Jaroslaw Kotowicz and Yatsuka Nakamura.

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