Abstract Algebra/Group Theory/Subgroup/Cyclic Subgroup/Euler's Totient Theorem

=Theorem= Let n be a positive integer. Let x be an integer relatively prime to n Let φ(n) = number of positive integers less than and relatively prime to n


 * $$x^{\varphi (n)} \equiv 1 \pmod{n}$$

=Proof= $$ \mathbb{Z}/n^{\times} $$ with multiplication mod n is a Group of positive integers less than and relatively prime to integer n.

φ(n) = o($$ \mathbb{Z}/n^{\times} $$)

Let X be the cyclic subgroup of $$ \mathbb{Z}/n^{\times} $$ generated by x mod n.

As X is subgroup of $$ \mathbb{Z}/n^{\times} $$
 * 0. o(X) divides o($$ \mathbb{Z}/n^{\times} $$)


 * 1. o($$ \mathbb{Z}/n^{\times} $$) / o(X) is an integer


 * 2. $$

x^{\varphi (n)} = x^{ o( \mathbb{Z}/n^{\times} ) } = x^{o(X) o(\mathbb{Z}/n^{\times}) / o(X)}

= [x ^ {o(X)} ]^{o( \mathbb{Z}/n^{\times}) / o(X)} = e^{o( \mathbb{Z}/n^{\times}) / o(X)} = e

$$