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Chinese Remainder

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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Chinese Remainder -

Tag(s) : Arithmetics

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Chinese Remainder

Chinese Remainder Calculator

 Display The smallest (positive) solution All solution in general form (if possible)

What is the Chinese Remainder Theorem? (Definition)

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? This exercise implies to calculate $x$ such that $x \equiv 2 \mod 3$ and $x \equiv 3 \mod 5$ and $x \equiv 2 \mod 7$

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:

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

How to calculate Chinese remainder?

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 A, modulo B) in equations of the form x = A mod B

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

When does the Chinese Remainder Theorem have no solution?

The system of equations with remainders $r_i$ and modulos $m_i$ has solutions only if the following modular equation is true: $$r_1 \mod d = r_2 \mod d = \cdots r_n \mod d$$ with $d$ the GCD of all modulos $m_i$.

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Chinese Remainder on dCode.fr [online website], retrieved on 2022-11-29, https://www.dcode.fr/chinese-remainder

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