Tool to compute congruences with the chinese remainder theorem. The Chinese Remainder Theorem helps to solve congruence equation systems in modular arithmetic.

Chinese Remainder - dCode

Tag(s) : Arithmetics

dCode is free and its tools are a valuable help in games, maths, geocaching, puzzles and problems to solve every day!

A suggestion ? a feedback ? a bug ? an idea ? *Write to dCode*!

Tool to compute congruences with the chinese remainder theorem. The Chinese Remainder Theorem helps to solve congruence equation systems in modular arithmetic.

The **Chinese remainder** theorem is the name given to a system of congruences (multiple simultaneous modular equations). The original problem is to calculate a number of elements which remainders (of their Euclidean division) are known.

__Example:__ If they are arranged by 3 there remains 2. If they are arranged by 5, there remain 3 and if they are arranged by 7, there remain 2. How many objects are there?

Take a list of $ k $ coprimes integers $ n_1, ..., n_k $ and their product $ n = \prod_{i=1}^k n_i $. For all integers $ a_1, ... , a_k $, it exists another integer $ x $ which is unique modulo $ n $, such as :

$$ \begin{matrix} x \equiv a_1\pmod{n_1} \\ \ldots \\ x \equiv a_k\pmod{n_k} \end{matrix} $$

To find a solution of the congruence system, take the numbers $ \hat{n}_i = \frac n{n_i} = n_1 \ldots n_{i-1}n_{i+1}\ldots n_k $ which are also coprimes. To find the modular inverses, use the Bezout theorem to find integers $ u_i $ and $ v_i $ such as $ u_i n_i + v_i \hat{n}_i = 1 $. Here, $ v_i $ is the modular inverse of $ \hat{n}_i $ modulo $ n_i $.

Take then the numbers $ e_i = v_i \hat{n}_i \equiv 1 \mod{n_i} $. A particular solution of the **Chinese remainders** theorem is $$ x = \sum_{i=1}^k a_i e_i~ $$

dCode accepts numbers as pairs (remainder, modulo) or written x = A mod B

__Example:__ $ (2,3),(3,5),(2,7) \iff \left\{ \begin{array}{ll} x = 2 \mod 3 \\ x = 3 \mod 5 \\ x = 2 \mod 7 \end{array} \right. \Rightarrow x = 23 $

dCode retains ownership of the source code of the script Chinese Remainder online. Except explicit open source licence (indicated Creative Commons / free), any algorithm, applet, snippet, software (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or any function (convert, solve, decrypt, encrypt, decipher, cipher, decode, code, translate) written in any informatic langauge (PHP, Java, C#, Python, Javascript, Matlab, etc.) which dCode owns rights will not be released for free. To download the online Chinese Remainder script for offline use on PC, iPhone or Android, ask for price quote on contact page !

chinese,remainder,theorem,modulo,congruence,equation,coprime,division

Source : https://www.dcode.fr/chinese-remainder

© 2020 dCode — The ultimate 'toolkit' to solve every games / riddles / geocaching / CTF.

Feedback

▲