Chinese Remainder Theorem

A theorem describing an isomorphism and solution criterion: when n factors as a product of pairwise coprime integers n1,...,nk, the ring Z/nZ is isomorphic to the product ∏ Z/niZ, and systems of congruences modulo the ni have a unique solution modulo n.

A theorem stating that a system of simultaneous congruences x ≡ a_i (mod m_i) with pairwise coprime moduli m_i has a unique solution modulo M = ∏ m_i; equivalently, the ring Z/MZ is isomorphic to the product of Z/m_iZ when the m_i are pairwise coprime.

A result that gives conditions for and a construction of simultaneous solutions to systems of congruences x ≡ a_i (mod m_i) when the moduli m_i are pairwise coprime; it asserts existence and uniqueness modulo the product M = ∏ m_i and provides an explicit reconstruction.