Chinese Remainder

Question 1

What is the Chinese Remainder Theorem? (Definition)

Answer

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:

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

Question 2

How to calculate Chinese remainder?

Answer

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 $

Question 3

When does the Chinese Remainder Theorem have no solution?

Answer

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 $.

Chinese Remainder