greatest common divisor of integers

整数的最大公因数

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

Other names / variants: gcd · greatest common factor · 最大公约数 · 最大公因子

Translation evidence: Literature usage

Meaning in context

For integers $a,b$ not both zero, the positive common divisor divisible by every common divisor; set $\gcd(0,0)=0$ by convention.

对不全为零的整数 $a,b$，能被每个公因数整除的正公因数；约定 $\gcd(0,0)=0$。

Example

$\gcd(252,198)=18$.

Usage and distinctions

In a general domain a gcd is only unique up to units; positivity selects a unique integer representative.

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Definition sources

1. Thomas W. Judson — Abstract Algebra: Theory and ApplicationsSection 2.2, gcd definition before Theorem 2.2.2; zero-zero convention is editorial · Checked 2026-10-14

Translation sources

1. 陈碧琴 — 矩阵初等变换在初等数论中的应用，南通工学院学报中文摘要与关键词，最大公因数; reference [5] title 最大公约数 · Checked 2026-10-14
2. 李文威 — 代数学方法 卷一，作者源文件唯一分解性，paragraph defining 最大公因子 before def:UFD · Checked 2026-10-14

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