CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-07-31

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Euler's totient is multiplicative: $\gcd (m, n) = 1$ implies $φ (m n) = φ (m) φ (n)$ for positive $m, n$

Statement

For positive integers $m, n$ with $\gcd (m, n) = 1$ ,

$φ (m n) = φ (m) φ (n) .$

Thus Euler's totient is multiplicative on coprime positive arguments.

Facts & Assumptions

Given: Coprime positive integers $m, n$ .

[L1]

The Chinese remainder map $Z / (m n) \to Z / m \times Z / n$ is a bijection preserving multiplication and $[1]$ (Chinese remainder theorem for a finite pairwise-coprime list: simultaneous residues determine one class modulo the product, and the resulting bijection preserves addition and multiplication).

[L2]

A unit is an element with a multiplicative inverse, and $φ (c)$ is the cardinality of the unit group modulo the positive integer $c$ (The unit group $(Z / n)^{\times}$ and Euler's totient $φ (n) = ∣ (Z / n)^{\times} ∣$ for $n \geq 1$ ).

[L3]

The Cartesian product of finite sets of cardinalities $r, s$ has cardinality $r s$ (The product rule: $∣ A \times B ∣ = ∣ A ∣ ∣ B ∣$ , and $∣ \prod_{i < m} A_{i} ∣ = \prod_{i < m} ∣ A_{i} ∣$ ).

Proof

technique · direct

1.1

A class $u$ modulo $m n$ is a unit if and only if both components of its CRT image are units: an inverse of $u$ maps to componentwise inverses, while inverses of both components form a pair whose unique CRT preimage multiplies with $u$ to the unique preimage of the identity.

L1L2

2.1

Consequently the CRT bijection restricts to a bijection $(Z / (m n))^{\times} \to (Z / m)^{\times} \times (Z / n)^{\times}$ . Taking finite cardinalities and applying [L3] gives $φ (m n) = φ (m) φ (n)$ .

step 1.1L2L3∎

Depends on

Used by

Dependency tree · two levels

27 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

K. Conrad, The Chinese Remainder Theorem (standard reference, not scraped)
Mathematics LibreTexts, Euler's phi Function (standard reference, not scraped)

Euler's totient is multiplicative: gcd⁡(m,n)=1gcd(m,n)=1 implies φ(mn)=φ(m)φ(n)φ(mn)=φ(m)φ(n) for positive m,nm,n