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

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

Answers to Questions

What is the Chinese Remainder Theorem ?

The original problem is to consider a number of elements which we know the remainder of their Euclidean division.

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?

Consider 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 :

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

How to calculate chinese remainder?

Software accepts numbers written in couple (remainder, modulus), but is may be easier to write lines of x = A MOD B

(2,3),(3,5),(2,7) => x = 23

x = 2 mod 3 x = 3 mod 5 x = 2 mod 7 => x = 23

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