Chinese Remainder Theorem Problems And Solutions Pdf

chinese remainder theorem problems and solutions pdf

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Chinese remainder theoremancient theorem that gives the conditions necessary for multiple equations to have a simultaneous integer solution.

For all integersaandb,the pair of congruencesx amodm, x bmodnhas a solution, and this solution is uniquely determined is important here is thatmandnare relatively prime. The Chinese remainder theorem says we can uniquely solve any pair of congruences that have relatively prime moduli.

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Chinese remainder theorem , ancient theorem that gives the conditions necessary for multiple equations to have a simultaneous integer solution. The theorem has its origin in the work of the 3rd-century- ad Chinese mathematician Sun Zi, although the complete theorem was first given in by Qin Jiushao.

The Chinese remainder theorem addresses the following type of problem. One is asked to find a number that leaves a remainder of 0 when divided by 5, remainder 6 when divided by 7, and remainder 10 when divided by The simplest solution is The theorem can be expressed in modern general terms using congruence notation. For an explanation of congruence, see modular arithmetic. Let n 1 , n 2 , …, n k be integers that are greater than one and pairwise relatively prime that is, the only common factor between any two of them is 1 , and let a 1 , a 2 , …, a k be any integers.

The theorem also gives a formula for finding a solution. Note that in the example above, 5, 7, and 12 n 1 , n 2 , and n 3 in congruence notation are relatively prime. There is not necessarily any solution to such a system of equations when the moduli are not pairwise relatively prime. Chinese remainder theorem Article Additional Info. Home Science Mathematics Chinese remainder theorem mathematics.

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External Websites. Some Materials for Discrete Mathematics. The Editors of Encyclopaedia Britannica Encyclopaedia Britannica's editors oversee subject areas in which they have extensive knowledge, whether from years of experience gained by working on that content or via study for an advanced degree See Article History. Learn More in these related Britannica articles: modular arithmetic. Modular arithmetic , in its most elementary form, arithmetic done with a count that resets itself to zero every time a certain whole number N greater than one, known as the modulus mod , has been reached.

Examples are a digital clock in…. He dealt with the case when moduli are relatively prime, and he then reduced the case when they are not by first eliminating common factors. Qin Jiushao , Chinese mathematician who developed a method of solving simultaneous linear congruences.

In Qin joined the army as captain of a territorial volunteer unit and helped quash a local rebellion. History at your fingertips. Sign up here to see what happened On This Day , every day in your inbox! Email address. By signing up, you agree to our Privacy Notice. Be on the lookout for your Britannica newsletter to get trusted stories delivered right to your inbox.

Chinese Remainder Theorem (The Definitive Guide)

Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up. I recall doing this same exact procedure when doing the egg problem figuring out how many eggs the lady started with, and it worked out fine. What am I doing wrong? The help would be appreciated.

Main menu Search. The Chinese Remainder Theorem. Here is one way to solve the problem. It remains to check that all such integers work. Whilst similar ideas to those used above will work here, it's getting a bit trickier.

Chinese remainder theorem , ancient theorem that gives the conditions necessary for multiple equations to have a simultaneous integer solution. The theorem has its origin in the work of the 3rd-century- ad Chinese mathematician Sun Zi, although the complete theorem was first given in by Qin Jiushao. The Chinese remainder theorem addresses the following type of problem. One is asked to find a number that leaves a remainder of 0 when divided by 5, remainder 6 when divided by 7, and remainder 10 when divided by The simplest solution is The theorem can be expressed in modern general terms using congruence notation. For an explanation of congruence, see modular arithmetic.

Category: Chinese remainder theorem word problems pdf

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This definitive guide covers proofs, examples, algorithms, applications, and the Chinese Remainder Theorem history. It also includes links to additional resources such as online articles, courses, books, and tutors to help students learn from various sources. Professionals can also use these resources to increase their knowledge of the field or help structure courses for their students.

Chinese Remainder Theorem Word Problems Pdf

ГЛАВА 29 Все еще нервничая из-за столкновения с Хейлом, Сьюзан вглядывалась в стеклянную стену Третьего узла. В шифровалке не было ни души. Хейл замолк, уставившись в свой компьютер. Она мечтала, чтобы он поскорее ушел. Сьюзан подумала, не позвонить ли ей Стратмору. Коммандер в два счета выставит Хейла - все-таки сегодня суббота. Но она отдавала себе отчет в том, что, если Хейла отправят домой, он сразу же заподозрит неладное, начнет обзванивать коллег-криптографов, спрашивать, что они об этом думают, В конце концов Сьюзан решила, что будет лучше, если Хейл останется.

 Н-нет… Не думаю… - Голос его дрожал. Беккер склонился над. - Вам плохо. Клушар едва заметно кивнул: - Просто… я переволновался, наверное.  - И замолчал.

 - Какого черта вы не позвонили Стратмору. - Мы позвонили! - не сдавалась Мидж.  - Он сказал, что у них все в порядке. Фонтейн стоял, тяжело дыша. - У нас нет причин ему не верить.

С какой стати университетский профессор… Это не университетские дела. Я позвоню и все объясню. Мне в самом деле пора идти, они связи, обещаю. - Дэвид! - крикнула .

 - Уничтожить всю нашу секретную информацию? - Сьюзан не могла поверить, что Танкадо совершит нападение на главный банк данных АНБ. Она перечитала его послание.

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Example 2. Find x such that 3x ≡ 6 (mod 12). Solution. Uh oh. This time we don't have a multiplicative inverse to.

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