integer congruence

整数同余

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

Other names / variants: congruence modulo an integer

Translation evidence: Literature usage

Meaning in context

For $m\ge1$, integers $a,b$ satisfy $a\equiv b\pmod m$ exactly when $m\mid(a-b)$.

对正整数 $m$，若 $m\mid(a-b)$，则称整数 $a,b$ 模 $m$ 同余。

Example

$23\equiv5\pmod6$ because $23-5=18$.

Usage and distinctions

Cancellation requires care: $2\cdot1\equiv2\cdot4\pmod6$ but $1\not\equiv4\pmod6$.

Read the encyclopedia explanation

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Keith Conrad — Modular Arithmetic, University of Connecticut author notesSection 2, Definition 2.1; Section 3, Theorem 3.8 on cancellation · 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三章，第一节 同余的概念及其基本性质 (PDF page 4); name evidence only · Checked 2026-10-15

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