21. Chinese Remainder Theorem chinese remainder theorem. Author hasinoff What is the chinese remainder theoremas it applies to solving equations involving the modulus operator? http://www.newton.dep.anl.gov/newton/askasci/1995/math/MATH056.HTM

Ask A Scientist Mathematics Archive Chinese remainder theorem Back to Mathematics Ask A Scientist Index NEWTON Homepage Ask A Question ... NEWTON is an electronic community for Science, Math, and Computer Science K-12 Educators. Argonne National Laboratory, Division of Educational Programs, Harold Myron, Ph.D., Division Director.

22. Chinese Remainder Theorem - Encyclopedia Article About Chinese Remainder Theorem encyclopedia article about chinese remainder theorem. chinese remainder theoremin Free online English dictionary, thesaurus and encyclopedia. http://encyclopedia.thefreedictionary.com/Chinese remainder theorem

Dictionaries: General Computing Medical Legal Encyclopedia Chinese remainder theorem Word: Word Starts with Ends with Definition The Chinese remainder theorem is any of a number of related results in abstract algebra Abstract algebra is the field of mathematics concerned with the study of algebraic structures such as groups, rings and fields. The term "abstract algebra" is used to distinguish the field from "elementary algebra" or "high school algebra" which teaches the correct rules for manipulating formulas and algebraic expressions involving real and complex numbers. Historically, algebraic structures usually arose first in some other field of mathematics, were specified axiomatically, and were then studied in their own right in abstract algebra. Because of this, abstract algebra has numerous fruitful connections to all other branches of mathematics. Click the link for more information. and number theory Traditionally, number theory is that branch of pure mathematics concerned with the properties of integers and contains many open problems that are easily understood even by non-mathematicians. More generally, the field has come to be concerned with a wider class of problems that arose naturally from the study of integers. Number theory may be subdivided into several fields according to the methods used and the questions investigated. See for example the list of number theory topics. Click the link for more information.

23. The Chinese Remainder Theorem The chinese remainder theorem. Theorem 1.7.31 (chinese remainder theorem) Letand be relatively prime integers. Then. (1.6). has a simultaneous solution . http://web.usna.navy.mil/~wdj/book/node37.html

Next: General Chinese remainder theorem Up: Congruences Previous: Second method (due to Contents Index The Chinese remainder theorem In this section, we see how to solve simple simultaneous congruences modulo . This will be applied to the study of the Euler -function. Theorem 1.7.31 (Chinese remainder theorem) Let and be relatively prime integers. Then has a simultaneous solution . Furthermore, if are two solutions to ( ) then Example 1.7.32 and proof if and only if , for some . Therefore, the truth of the existence claim above is reduced to finding an integer such that . Since , there are integers such that , so . This implies , where . Thus a solution exists. To prove uniqueness , let and . Subtracting, we get and . Since , the result follows. Subsections General Chinese remainder theorem Next: General Chinese remainder theorem Up: Congruences Previous: Second method (due to Contents Index David Joyner 2002-08-23

24. General Chinese Remainder Theorem General chinese remainder theorem. Theorem 1.7.33 (chinese remainder theorem,general version) Let be pairwise relatively prime integers. Let . Then. (1.7). http://web.usna.navy.mil/~wdj/book/node38.html

Next: An application to Euler's Up: The Chinese remainder theorem Previous: The Chinese remainder theorem Contents Index General Chinese remainder theorem Theorem 1.7.33 (Chinese remainder theorem, general version) Let be pairwise relatively prime integers. Let . Then has a simultaneous solution . Furthermore, if are two solutions to ( ) then This follows from the case proven above using mathematical induction. The details are left as an exercise. We give a different proof below. proof : As runs over all integers , the -tuples form a collection of distinct -tuples in . (Exercise: show why they are distinct.) On the other hand, there are distinct, -tuples with . Therefore, each -tuple must equal one of the , for David Joyner 2002-08-23

25. Making Mathematics: Mathematics Tools: The Chinese Remainder Theorem Home Mathematics Tools The chinese remainder theorem. The ChineseRemainder Theorem. The chinese remainder theorem states that http://www2.edc.org/makingmath/mathtools/remainder/remainder.asp

Home Mathematics Tools The Chinese Remainder Theorem The Chinese Remainder Theorem states that for relatively prime m , m , ... , there is a unique solution (mod m m ...) to the system of congruences x = a (mod m x = a (mod m For a discussion and some sample problems, see Chinese Remainder Theorem Making Mathematics Home Mathematics Projects Students ... Parents Translations of mathematical formulas for web display were created by Webmaster: Terry Dash at tdash@edc.org

26. PlanetMath Chinese Remainder Theorem chinese remainder theorem, (Theorem). Suppose we have a set of congruencesof the form. Attachments chinese remainder theorem proof (Proof) by vampyr. http://planetmath.org/encyclopedia/ChineseRemainderTheorem.html

27. PlanetMath Chinese Remainder Theorem chinese remainder theorem, (Theorem). Let be a commutative ring with identity. Attachmentsproof of chinese remainder theorem (Proof) by mclase. http://planetmath.org/encyclopedia/ChineseRemainderTheorem2.html

28. Chinese Remainder Theorem :: Online Encyclopedia :: Information Genius chinese remainder theorem. One of the most general forms of the Chinese remaindertheorem can be formulated for rings and (twosided) ideals. http://www.informationgenius.com/encyclopedia/c/ch/chinese_remainder_theorem.htm

Quantum Physics Pampered Chef Paintball Guns Cell Phone Reviews ... Science Articles Chinese remainder theorem Online Encyclopedia 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

29. Chinese Remainder Theorem chinese remainder theorem. Fatal error Call to undefined function encode_cyr()in /afs/unibonn.de/home/manfear/public_php/mathdict-entry.php on line 51 http://www.uni-bonn.de/~manfear/mathdict-entry.php?term=chinese remainder theore

30. ChineseRemainderTheorem chinese remainder theorem (English). Search for Chinese remaindertheorem in NRICH PLUS maths.org Google. Definition level 3. http://thesaurus.maths.org/mmkb/entry.html?action=entryByConcept&id=2304&langcod

31. The Chinese Remainder Theorem The chinese remainder theorem. Recall each . In terms of rings, the ChineseRemainder Theorem asserts that the natural map. is an isomorphism. http://modular.fas.harvard.edu/papers/ant/html/node31.html

Next: Discrimannts, Norms, and Finiteness Up: Chinese Remainder Theorem Previous: Chinese Remainder Theorem Contents Index The Chinese Remainder Theorem Recall that the Chinese Remainder Theorem from elementary number theory asserts that if are integers that are coprime in pairs, and are integers, then there exists an integer such that for each . In terms of rings, the Chinese Remainder Theorem asserts that the natural map is an isomorphism. This result generalizes to rings of integers of number fields. Lemma If and are coprime ideals in , then Proof . The ideal is the largest ideal of that is divisible by (contained in) both and . Since and are coprime, is divisible by , i.e., . By definition of ideal , which completes the proof. Remark This lemma is true for any ring and ideals such that . For the general proof, choose and such that . If then so , and the other inclusion is obvious by definition. Theorem (Chinese Remainder Theorem) Suppose are ideals of such that for any . Then the natural homomorphism induces an isomorphism Thus given any then there exists such that for Proof . First assume that we know the theorem in the case when the are powers of prime ideals. Then we can deduce the general case by noting that each

32. Chinese Remainder Theorem chinese remainder theorem. In this section chapter very well. SubsectionsThe chinese remainder theorem. William Stein 200405-06. http://modular.fas.harvard.edu/papers/ant/html/node30.html

Next: The Chinese Remainder Theorem Up: Classical Viewpoint Previous: Essential Discriminant Divisors Contents Index Chinese Remainder Theorem In this section we will prove the Chinese Remainder Theorem for rings of integers, deduce several surprising and useful consequences, then learn about discriminants, and finally norms of ideals. We will also define the class group of and state the main theorem about it. The tools we develop here illustrate the power of what we have already proved about rings of integers, and will be used over and over again to prove other deeper results in algebraic number theory. It is essentially to understand everything we discuss in this chapter very well. Subsections The Chinese Remainder Theorem William Stein 2004-05-06

33. Chinese Remainder Theorem Modular Mathematics, chinese remainder theorem. Search Site map Contactus Join our mailing list Books chinese remainder theorem. http://www.mathreference.com/num-mod,chr.html

Modular Mathematics, Chinese Remainder Theorem Search Site map Contact us Join our mailing list ... Books Main Page Numbers Modular Mathematics Use the arrows at the bottom to step through Modular Math. Chinese Remainder Theorem Given a set of values c c ... c n , and a set of mutually coprime moduli m m ... m n , is there an integer x such that x = c i mod m i for each i in 1 through n? Let z be the product of all the moduli. If x is a solution then so is x plus any multiple of z. If w is not a multiple of z, say w is not divisible by m , then x+w will not equal c mod m , and x+w will not be a solution. The solution, if it exists, is well defined mod z. To show that a solution exists, we simply construct one. Let a i be the product of all the moduli other than m i . Verify that a i and m i are coprime. Let b i be the inverse of a i mod m i . Finally, let x be the sum of a i b i c i for all i in 1 ... n. Verify that x satisfies all n equations simultaneously. If the original moduli are not coprime, split each equation up into a set of equations by factoring the composite modulus into prime powers. Then consider all the equations together. Equations sharing a common prime modulus are either redundant or inconsistent. An inconsistent example is x = 4 mod 6 and x = 11 mod 15 . This would force x = 1 mod 3 and x = 2 mod 3, which is impossible. The example x = 4 mod 6 and x = 7 mod 15 has the solution x = 22 mod 30.

34. Chinese Remainder Theorem Rings, chinese remainder theorem. Search Site map Contact us Join ourmailing list Books Back to Theory. chinese remainder theorem. The http://www.mathreference.com/ring,chr.html

Rings, Chinese Remainder Theorem Search Site map Contact us Join our mailing list ... Books Main Page Rings Use the arrows at the bottom to step through Ring Theory. Chinese Remainder Theorem The chinese remainder theorem was developed for modular arithmetic , but it generalizes to ideals in a commutative ring r. Let h h ... h n be a set of coprime ideals. By coprime, we mean the sum of any two ideals spans the ring. Let j be the product of all these ideals. We will prove r/j is isomorphic to the direct product of the quotient rings r/h i , as i runs from 1 to n. An element in r/j can be mapped to the ith component in the direct product via r/h i . This is a well defined ring homomorphism, since each h i wholly contains j. We need to show it is 1-1 and onto. Focus on h . We know x + y = 1 for some x in h and y in h . Do the same for each h i in the set. Multiply all these equations together, and something in h + something in the product of the other ideals gives 1. Write this as x + y = 1. Reduce mod h , and y = -1. If y

35. CHINESE REMAINDER THEOREM chinese remainder theorem. Let hcf (n1,n2,n3, ,nr)=1. Then the system of linearcongruences xa1 mod n1. xa2 mod n2. xa3 mod n3  . xar mod nr. http://www.bearnol.pwp.blueyonder.co.uk/Math/chinese.htm

CHINESE REMAINDER THEOREM Let hcf (n1,n2,n3, ,nr)=1. Then the system of linear congruences: has a simultaneous solution, unique modulo n1n2n3 nr Proof: Let n=n1n2n3 nr Let X=a1N1x1 + a2N2x2 +  + arNrxr so X == akNkxk == ak.1 == ak [mod nk] Suppose Y is another solution:

36. Hilbert Functions And The Chinese Remainder Theorem: Open Problems Hilbert Functions and the chinese remainder theorem Open Problems. Notes This20 minute talk was given February 27, 1999, at the UNL Regional Workshop. http://www.math.unl.edu/~bharbour/UNLregionwstalk.html

Hilbert Functions and the Chinese Remainder Theorem: Open Problems Notes: This 20 minute talk was given February 27, 1999, at the UNL Regional Workshop. For this talk let k be any algebraically closed field. Preliminary Problems Problem 1 : Given points x , ... , x n of k and values v , ... , v n of k, find all polynomials f(x) in k[x] such that f(x i ) = v i for all i. Solution : There exists a unique solution, f L (x), of degree at most n-1; it is given by the Lagrange interpolation formula: Let g(x) = (x-x )...(x-x n Let g i (x) = g(x)/(x-x i ) [This is a polynomial.] Then f L (x) = (v /g (x ))g (x) + ... + (v n /g n (x n ))g n (x) The complete set of solutions is f L (x) + (g); i.e., f L (x) + p(x)g(x), where p(x) is any polynomial. [In the usual notation, (g) is the ideal generated by g(x).] Restatement of Problem 1 and Solution : The problem is to find all f(x) conguent mod I j to v j , for all j, where I j is the ideal (x-x j ) of the ring R=k[x]. I.e., given an element v = (v , ... , v n ) of R/I x ... x R/I n , find all elements f of R mapping to v under the homomorphism H : f -> (f , ... , f

37. Chinese Remainder Theorem - InformationBlast chinese remainder theorem Information Blast. chinese remainder theorem.The chinese remainder theorem is any of a number of related http://www.informationblast.com/Chinese_remainder_theorem.html

Chinese remainder theorem The Chinese remainder theorem is any of a number of related results in abstract algebra and number theory 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 e i s n n i , then we have e i mod n i e i mod n j j i The number x i k a i e i then solves the given system of simultaneous congruences. For example, consider the problem of finding an integer x such that x mod x mod x mod Using the extended Euclidean algorithm for 3 and 4×5 = 20, we find (-13) × 3 + 2 × 20 = 1 (i.e.

38. Solving Congruences: The Chinese Remainder Theorem Solving Congruences The chinese remainder theorem. This is done by the Chinese RemainderTheorem, socalled because it appeared in ancient Chinese manuscripts. 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

39. Chinese Remainder Theorem Corollary chinese remainder theorem Corollary. I don t have the proof here, but I ll getaround to writing it up. Anyway, the chinese remainder theorem is Theorem. http://begghilos2.ath.cx/~jyseto/Academia/Math-CRTC.html

40. Chinese Remainder Theorem Back to the Table of Contents chinese remainder theorem. How many things are there? .This is the first known work involving the chinese remainder theorem (CRT). http://www.andrews.edu/~calkins/math/biograph/199899/topcrt.htm

Back to the Table of Contents 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. 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?" This is the first known work involving the Chinese Remainder Theorem (CRT). It is from the book "Sun Tzu Suan Ching" ("Master Sun's Mathematical Manual") written by Sun Zi (also called Master Sun). Sun Zi was probably a Buddhist scholar or monk. This is the only problem in the book relating to the CRT, so we don't know if he made a general method to solve these types of problems. There is some dispute about when problem 26 (up above) was written. Some experts say "Sun Tzu Suan Ching" was written in the late 3rd century BC, others argue it was created in the 4th century. Although little was known about Sun Zi or when he lived, he is considered an important part of mathematical history. The Chinese Remainder Theorem is: If m , m , m . . . m

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