division algorithm for integers

整数带余数除法

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

Other names / variants: division with remainder

Translation evidence: Literature usage

Meaning in context

For a∈Za\in\mathbb Z and b>0b>0, there are unique integers q,rq,r with a=bq+ra=bq+r and 0≤r<b0\le r<b.

对整数 aa 和正整数 bb，存在唯一整数 q,rq,r 使 a=bq+ra=bq+r 且 0≤r<b0\le r<b。

Example

−17=5(−4)+3-17=5(-4)+3 has quotient −4-4 and remainder 33.

Usage and distinctions

The division theorem specifies one quotient and remainder; the Euclidean algorithm repeatedly applies it to compute a gcd.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

  1. Thomas W. Judson — Abstract Algebra: Theory and ApplicationsSection 2.2, Theorem 2.2.1 · Checked 2026-10-14

Translation sources

  1. 新乡学院数学与统计学院 — 初等数论课程教学大纲第一章，第一节，第3项 带余数除法定理 (PDF page 2); name evidence only · Checked 2026-10-15
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