Euler's theorem on congruences

欧拉定理

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

Translation evidence: Literature usage

Meaning in context

If $n\ge2$ and $\gcd(a,n)=1$, then $a^{\varphi(n)}\equiv1\pmod n$.

若 $n\ge2$ 且 $\gcd(a,n)=1$，则 $a^{\varphi(n)}\equiv1\pmod n$。

Example

$3^4=81\equiv1\pmod{10}$ since $\varphi(10)=4$.

Usage and distinctions

Coprimality cannot be weakened to $a\not\equiv0$: $2^4\not\equiv1\pmod{10}$.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Keith Conrad — Euler's Theorem, University of Connecticut author notesSection 1, Theorem 1.2 · Checked 2026-10-15

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，第四节 欧拉定理、费尔马定理 (PDF page 4); name evidence only · Checked 2026-10-15

Original encyclopedia and terminology content: CC BY-SA 4.0. Contribute or report a correction.