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

Statement

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

$\varphi(mn)=\varphi(m)\varphi(n).$

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

Facts & Assumptions

Given: Coprime positive integers $m,n$ .

[L1]

The Chinese remainder map $\mathbb Z/(mn)\to\mathbb Z/m\times\mathbb Z/n$ 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 $\varphi(c)$ is the cardinality of the unit group modulo the positive integer $c$ (The unit group $(\mathbb{Z}/n)^\times$ and Euler's totient $\varphi(n)=\lvert(\mathbb{Z}/n)^\times\rvert$ for $n\ge1$ ).

[L3]

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

Proof

technique · direct

1.1

A class $u$ modulo $mn$ 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 $(\mathbb Z/(mn))^\times\to(\mathbb Z/m)^\times\times(\mathbb Z/n)^\times$ . Taking finite cardinalities and applying [L3] gives $\varphi(mn)=\varphi(m)\varphi(n)$ .

step 1.1L2L3∎

Depends on

Used by

Euler's product formula φ(n)=n∏_p∣ n(1-1/p)=∏_pᵏ∥ n(pᵏ-pᵏ⁻¹) for n≥1, stated through a finite injective list of its prime divisors Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 82 results over 19 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click 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)=1\gcd(m,n)=1gcd(m,n)=1 implies φ(mn)=φ(m)φ(n)\varphi(mn)=\varphi(m)\varphi(n)φ(mn)=φ(m)φ(n) for positive m,nm,nm,n