# Euler-phi function is multiplicative if coprime

This article is about a fact in number theory that has a direct proof using group theory.

## Statement

### Verbal statement

The Euler-phi function is multiplicative on coprime natural numbers.

### Statement with symbols

For any natural numbers that are coprime to each other, .

## Proof

**Given**:

coprime positive integers

**To prove**:
**Proof**: Consider a finite cyclic group G of order . Denote by the set of all elements in G that have orders respectively. We will show that .

Define a map by . Since the order of and are coprime, and is abelian, their product must have order , so this map is well-defined. If then , or equivalently, . They are elements of subgroups of coprime orders that must intersect trivially. So, and , the map is injective. Thus, .

Define another map by . If then and . Suppose . Then . Therefore . Similarly, . So, and must have a common divisor which is the order of . Thus, . Hence our map is again injective, so .

Therefore, , as desired.