Search results with tag "Chinese remainder theorem"
The Chinese Remainder Theorem - University of Illinois at ...
homepages.math.uic.eduChinese Remainder Theorem tells us that there is a unique solution modulo m, where m = 11 ⋅ 16 ⋅ 21 ⋅ 25 = 92400. We apply the technique of the Chinese Remainder Theorem with k = 4, m 1 = 11, m 2 = 16, m 3 = 21, m 4 = 25, a 1 = 6, a 2 = 13, a 3 = 9, a 4 = 19, to obtain the solution. We compute z 1 = m / m 1 = m 2 m 3 m 4 = 16 ⋅ 21 ⋅ ...
Math 127: Chinese Remainder Theorem - CMU
www.math.cmu.eduExample 5. Use the Chinese Remainder Theorem to nd an x such that x 2 (mod5) x 3 (mod7) x 10 (mod11) Solution. Set N = 5 7 11 = 385. Following the notation of the theorem, we have m 1 = N=5 = 77, m 2 = N=7 = 55, and m 3 = N=11 = 35. We now seek a multiplicative inverse for each m i modulo n i. First: m 1 77 2 (mod5), and hence an inverse to m 1 ...
THE CHINESE REMAINDER THEOREM
kconrad.math.uconn.eduThe Chinese remainder theorem can be extended from two congruences to an arbitrary nite number of congruences, but we have to be careful about the way in which the moduli are relatively prime. Consider the three congruences x 1 mod 6; x 4 mod 10; x 7 mod 15: While there is no common factor of 6, 10, and 15 greater than 1, these congruences do
Section 4.3 - The Chinese Remainder Theorem
zimmer.csufresno.eduSolution: Since 8 and 9 are relatively prime, we can use the Chinese remainder theorem to solve the congruences x ≡ 1 (mod 8) x ≡ 3 (mod 9) One comes up with x ≡ 57 (mod 72). Thus since 12 divides 72, we must also have x ≡ 57 (mod 12). But 57 6≡2 (mod 12)
Historical development of the Chinese remainder theorem
www.math.harvard.eduThe Chinese Remainder Theorem 291 where a, b, c are natural numbers, was the same as the congruence ax ~- b (mod c). Therefore the system of congruences in Example 2 may be converted into 100x ~ 32 (mod 83) ~ 70 (rood 110) ~ 30 (mod 135), and that in …
The Chinese Remainder Theorem
gauss.math.luc.eduThe Chinese Remainder Theorem We now know how to solve a single linear congruence. In this lecture we consider how to solve systems of simultaneous linear congruences.
Chapter 8: Fast Convolution
people.ece.umn.eduChinese remainder theorem) ... • The application of Lagrange interpolation theorem into linear convolution Consider an N-point sequence h = {h 0 ,h 1,..., h N −1} and an L-point sequence x = {x 0 , x 1,..., x L −1}. The linear convolution of h and x can be expressed in terms of polynomial
The Chinese Remainder Theorem - UC Denver
www-math.ucdenver.eduby 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 ≡ 0 mod 2 and x ≡ 0 mod 5.
The number of homomorphisms from Z to Z
users.metu.edu.trKeywords and phrases : Homomorphisms, groups, rings, Chinese Remainder Theorem. 2010 Mathematics Subject Classification : 11A07 1 Introduction In order to determine the number of homomorphisms, we do not need to assume previous knowledge from group theory or ring theory, except for the de nition of group and ring homomorphism. With respect to
Chapter 4.4: Systems of Congruences
math.berkeley.eduChinese Remainder Theorem Decide whether the system has a solution. If it does, nd it. 1. x 3 (mod 8), x 1 (mod 7) Try x = 8a+7b. mod 8, we get 3 x 7b (mod 8), and solving gives b = 5. mod 7, we get 1 x 8a (mod 7), so a 1 (mod 7). Therefore one …
Math 110 Homework 3 Solutions
www.math.lsa.umich.eduBy the Chinese Remainder Theorem, we already knew that the map (Z=mnZ) ! (Z=mZ) (Z=nZ) c(mod nm) 7! c(mod m) ;c(mod n) is one-to-one and onto. Now we know that it maps every unit c(mod mn) to a pair of units c(mod m) ;c(mod n) , and conversely that every pair of units is in the image of a unit c(mod mn). We conclude that the map re-