61. Chinese Remainder Theorem And The Chinese Problem Mathematics Simultaneous Congruence Equations Number Theory Chinese Remainder Theorem and the Chinese Problem Number Theory Contents http://www.trans4mind.com/personal_development/mathematics/numberTheory/chineseR

Ken Ward's Mathematics Pages Number Theory Chinese Remainder Theorem and the Chinese Problem Number Theory Contents Page Contents Chinese Problem Chinese Remainder Theorem Solving the Problem Using Successive Substitution Rationale Chinese Problem The Chinese Problem was first introduced by Sun Tsu (460 AD), and problems of that kind are often referrred to as the Chinese Problem:Oystein Ore mentions the following problem due to Brahmagupta (598) When the eggs are taken out of a basket 2, 3, 4, 5, 6, 7 at a time, the remainders 1, 2, 3, 4, 5 and 0. How many eggs were in the basket? Apparently such problems amused both the high and the low at that time. The divisors are not relatively prime, and relative primality is not a condition for solvability. Chinese Remainder Theorem The Chinese Remainder Theorem is a statement of the conditions under which a set of simultaneous congrent equations is solvable. We will answer this question later. Solving the Problem Using Successive Substitution We have already used a method to solve think kind of problem. The following "low-tech" method is more efficient, however. When the eggs were taken out 2 at a time, the remainder was 1. We can write an equation:

62. The Chinese Remainder Theorem Aug 7, 1995 The Chinese Remainder Theorem (CRT) gives the answer to the problem Find the number x, that satisfies all the n equations simultaneously 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

63. Chinese Remainder Theorem Definition of Chinese remainder theorem, possibly with links to more information and implementations. http://xw2k.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 Entry modified 17 December 2004. HTML page formatted Mon Sep 27 10:31:22 2010. Cite this as: Paul E. Black, "Chinese remainder theorem", in Dictionary of Algorithms and Data Structures [online], Paul E. Black, ed., U.S. National Institute of Standards and Technology . 17 December 2004. (accessed TODAY) Available from: http://xw2k.nist.gov/dads/HTML/chineseRmndr.html

64. RSA Speedup With Chinese Remainder Theorem Immune Against Hardware File Format PDF/Adobe Acrobat Quick View http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.122.9916&rep=rep1&a

65. Modular Arithmetic File Format PDF/Adobe Acrobat Quick View http://www.cs.cmu.edu/~adamchik/21-127/lectures/congruences_print.pdf

66. Chinese Remainder Theorem | Facebook Welcome to the Facebook Community Page about Chinese remainder theorem, a collection of shared knowledge concerning Chinese remainder theorem. http://www.facebook.com/pages/Chinese-remainder-theorem/106339806071525

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67. Section 7.1: Solving Two Congruences 7.1 Solving Two Congruences. The Chinese Remainder Theorem is one of the oldest theorems in number theory. Your first job is to discover the right statement of this theorem. http://www.math.mtu.edu/mathlab/COURSES/holt/dnt/chinese1.html

7.1 Solving Two Congruences The Chinese Remainder Theorem is one of the oldest theorems in number theory. Your first job is to discover the right statement of this theorem. Most of the statement of the theorem is provided in the next section - you just need to fill in the missing part. 7.1.1 The Chinese Remainder Theorem for Two Congruences Below is an incomplete statement of the Chinese Remainder Theorem for two congruences. Chinese Remainder Theorem : If m and m are positive integers such that , then for any integers a and a , the pair of congruences x a (mod m ) and x a (mod m has a unique solution x modulo m m To complete Research Question 1, you need to figure out what condition is required in place of the above to make the statement true. Notice that the placement occurs after the introduction of m and m but before a and a . This means that the missing condition should involve m and m but not a or a . So to complete the statement of the theorem, you need to determine what condition on m and m allows one to find a unique solution to the congruences for all choices of a and a Research Question 1 Complete the statement of the Chinese Remainder Theorem for two congruences.

68. Math Forum - Ask Dr. Math Jul 25, 2001 Professor Carroll tries to divide his class into three groups, but two students are left http://mathforum.org/dr.math/problems/donald.07.21.01.html

Associated Topics Dr. Math Home Search Dr. Math Chinese Remainder Theorem and Modular Arithmetic Date: 07/25/2001 at 13:34:55 From: Donald Subject: Chinese Remainder Theorem I was wondering if there is any way to solve this using the Chinese Remainder Theorem rather than just trial and error. Professor Carroll tries to divide his class into three equal groups, but two students do not have a group. When he tries to divide them into five groups, three people are left. Finally, a configuration of seven equal groups leaves two people hanging. What is the smallest possible number of students in Professor Caroll's class? Thank you for your help. http://mathforum.org/dr.math/problems/prufrock.4.25.00.html http://mathforum.org/dr.math/ Associated Topics High School Number Theory Search the Dr. Math Library: Find items containing (put spaces between keywords): Click only once for faster results: [ Choose "whole words" when searching for a word like age. all keywords, in any order at least one, that exact phrase parts of words whole words Submit your own question to Dr. Math

69. Summer High School 2009 Aaron Bertram 5. The Chinese Remainder File Format PDF/Adobe Acrobat Quick View http://www.math.utah.edu/~bertram/HighSchool/5Chinese.pdf

70. Chinese Remainder Theorem A Mechanical Proof of the Chinese Remainder Theorem David M. Russinoff. This paper (ps, pdf), which was presented at ACL2 Workshop 2000 (see slides ps, pdf), describes an ACL2 http://www.russinoff.com/papers/crt.html

A Mechanical Proof of the Chinese Remainder Theorem David M. Russinoff This paper ( ps pdf ), which was presented at ACL2 Workshop 2000 (see slides: ps pdf ), describes an ACL2 proof of the Chinese Remainder Theorem: If m m k are pairwise relatively prime moduli and a a k are natural numbers, then there exists a natural number x that simultaneously satisfies x a i (mod m i i k The entire proof is contained in the single event file crt.lisp , except that it depends on some lemmas from the author's library of floating-point arithmetic . In order to certify this file (after obtaining and certifying the library), first replace each of the two occurrences of " /u/druss/ " with the path to the directory under which your copy of the library resides. A second event file, summary.lisp , which contains the definitions and main lemmas involved in the proof, may then be certified.

71. MODULAR EXPONENTIATION VIA THE EXPLICIT CHINESE REMAINDER THEOREM File Format PDF/Adobe Acrobat Quick View http://cr.yp.to/antiforgery/meecrt-20060914-ams.pdf

72. Chinese Remainder Theorem Pascal Paillier, Publickey cryptosystems based on composite degree residuosity classes, Proceedings of the 17th international conference on Theory and application of cryptographic http://portal.acm.org/citation.cfm?id=260951

73. Math 126 Number Theory File Format PDF/Adobe Acrobat Quick View http://aleph0.clarku.edu/~djoyce/ma126/meeting17.pdf

74. Chinese Remainder Theorem HTML Page Millersville University/Department of Mathematics Chinese Remainder Theorem Applet. The Chinese Remainder Theorem is an ancient result taking its name from the work entitled http://banach.millersville.edu/~bob/math478/ChineseRemainder.html

Millersville University/Department of Mathematics Chinese Remainder Theorem Applet The Chinese Remainder Theorem is an ancient result taking its name from the work entitled Suangching by Sun Tsu. It is used to solve a system of linear congruences. In the left-hand column of textfields below fill in the remainders obtained when mod-ing by the relatively prime positive integers in the textfields in the right-hand column. Press the Solve button to find the solution. If you find a case in which the applet fails to function or gives erroneous results, please send me the values of the remainders and divisors which you entered and the contents of your Java console. My modest goal is to have this applet run reliably and accurately. You need a Java-enabled browser running JDK 1.1.x or greater to run this applet. Page maintained by: Bob Buchanan Bob.Buchanan@millersville.edu Last updated:

75. S.O.S. Mathematics CyberBoard :: View Topic - Chinese Remainder Theorem 3 posts 3 authors - Last post May 3Since , the Chinese Remainder Theorem is not applicable. Mathematicians are like lovers. Grant a mathematician the least principle, http://www.sosmath.com/CBB/viewtopic.php?p=204036&sid=b9a59bfc7faccadbade821

76. The Chinese Remainder Theorem (the CRT) The Chinese Remainder Theorem (the CRT) Brain Teasers discussion http://www.physicsforums.com/showthread.php?t=243035

77. COMPUTING IGUSA CLASS POLYNOMIALS VIA THE CHINESE REMAINDER File Format PDF/Adobe Acrobat Quick View http://research.microsoft.com/en-us/um/people/klauter/ELMay10.pdf

78. Chinese Remainder Theorem Due To Gauss Mathematics This formula is what is normally called the Chinese Remainder Theorem. http://www.trans4mind.com/personal_development/mathematics/numberTheory/chineseR

Ken Ward's Mathematics Pages Number Theory Chinese Remainder Theorem due to Gauss Number Theory Contents Page Contents Chinese Remainder Theorem due to Gauss Example 1 Example 2 Proof of the Chinese Remainder Theorem ... The solution is unique modulo M Chinese Remainder Theorem due to Gauss We seek to solve the set of equations: x (mod m x (mod m x n n (mod m n ) [1.1.n] Where the m's are relatively prime. This is a condition for the use of the formula below (called the Chinese Remainder Theorem) although, as we know, the Chinese Problem might be solvable anyway. Let M=product of the m's=m ...m n Let M i =M/m i , where i=1...n, that is it is the product of all the m's except m i Also The formula called the Chinese Remainder Theorem is: Example 1 A number when divided by 3, gives a remainder 2; when divided by 4, a remainder 3 and when divided by 5, a remainder 4. We seek the number (least positive number). k m k M k Inverse r k r *M *inv Provided the divisors are relatively prime, the formula can be used to find a solution. M=60 The number is therefore 59. Example 2 A number when divided by 7, gives a remainder 5; when divided by 9, a remainder 2 and when divided by 11, a remainder 10. We seek the number (least positive number).

79. A Mechanical Proof Of The Chinese Remainder Theorem David M. Russino File Format Adobe PostScript View as HTML http://www.cs.utexas.edu/users/moore/acl2/v4-1/distrib/acl2-sources/books/worksh

80. THEORY OF NUMBERS Computations (1) Use The Chinese Remainder File Format PDF/Adobe Acrobat Quick View http://www.cims.nyu.edu/~burr/Classes/Homework3Solutions.pdf

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