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The Chinese Remainder Theorem (CRT) is a fundamental result in number theory that provides a unique solution to a system of simultaneous linear congruences with pairwise coprime moduli. It states that if you have several congruences, each with a different modulus, and these moduli are pairwise coprime, then there exists a unique solution modulo the product of these moduli.
Statement of the Theorem
Given pairwise coprime positive integers ( n_1, n_2, \ldots, n_k ) and arbitrary integers ( a_1, a_2, \ldots, a_k ), the system of simultaneous congruences: [ x \equiv a_1 \pmod{n_1} ] [ x \equiv a_2 \pmod{n_2} ] [ \vdots ] [ x \equiv a_k \pmod{n_k} ] has a unique solution modulo ( N = n_1 \cdot n_2 \cdots n_k ) 12.
Proof and Construction
To find a solution to the system of congruences using the Chinese Remainder Theorem, follow these steps:
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