Chinese remainder theorem for integers

整数中国剩余定理

Number Theory · Undergraduate · Mathematics Resources editorial compilation (AI-assisted)

Other names / variants: CRT

Translation evidence: Literature usage

Meaning in context

For pairwise coprime positive integers m1,…,mrm_1,\ldots,m_r, any prescribed residues have a simultaneous integer solution unique modulo ∏imi\prod_i m_i.

对两两互素的正整数 m1,…,mrm_1,\ldots,m_r，任意指定的剩余类都有同时满足的整数解，且解模 ∏imi\prod_i m_i 唯一。

Example

x≡2(mod3), x≡3(mod5)x\equiv2\pmod3,\ x\equiv3\pmod5 gives x≡8(mod15)x\equiv8\pmod{15}.

Usage and distinctions

For noncoprime moduli, compatibility is required and the uniqueness modulus becomes the least common multiple.

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Recommended reading in this subject · Preparation & skip guidance

Definition sources

  1. 李文威 — 代数学方法 卷一，第五章，作者源文件Theorem prop:CRT and following paragraph specializing R=Z, I_i=a_i Z · Checked 2026-10-15

Translation sources

  1. 李文威 — 代数学方法 卷一，第五章，作者源文件定理 prop:CRT 中国剩余定理，紧随的初等数论特例 · Checked 2026-10-15
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