 ##  [Chinese Remainder Theorem](/chinese-remainder-theorem) 

  ##  [Chinese Remainder Theorem](https://mathlogic.quantumdictionary.io/chinese-remainder-theorem-0) 

  
 [![Mathematics & Logic Dictionary](/sites/default/files/styles/large/public/2026-01/Mathematics%20%26%20Logic.png.webp?itok=UhtTRPnp)](/topic-specific-dictionaries/natural-formal-sciences/mathematics-logic)

- Natural &amp; Formal Sciences -

**Mathematics &amp; Logic Dictionary**

 
Definition

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.

 
 ##  [Chinese Remainder Theorem](https://puremath.quantumdictionary.io/chinese-remainder-theorem-1) 

  
 [![Pure Mathematics Dictionary](/sites/default/files/styles/large/public/2026-01/Pure%20Mathematics.png.webp?itok=5pZnFQ59)](/topic-specific-dictionaries/mathematics-logic/pure-mathematics)

- Mathematics &amp; Logic -

**Pure Mathematics Dictionary**

 
Definition

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.

 
 ##  [Chinese Remainder Theorem](https://algebra.quantumdictionary.io/chinese-remainder-theorem-2) 

  
 [![Algebra](/sites/default/files/styles/large/public/2026-01/Algebra.png.webp?itok=3pHxBnUF)](/topic-specific-dictionaries/pure-mathematics/algebra)

- Pure Mathematics -

**Algebra Dictionary**

 
Definition

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.