Euler's Theorem

Theorem
Let $a, m \in \Z$ be coprime integers: $a \perp m$.

Let $\map \phi m$ be the Euler $\phi$ function of $m$.

Then:
 * $a^{\map \phi m} \equiv 1 \pmod m$

Proof
Let $\eqclass a m$ denote the residue class modulo $m$ of $a$.

Since $a \perp m$, it follows by Reduced Residue System under Multiplication forms Abelian Group that $\eqclass a m$ belongs to the abelian group $\struct {\Z'_m, \times}$.

Let $k = \order {\eqclass a m}$ where $\order {\, \cdot \,}$ denotes the order of a group element.

By Order of Element Divides Order of Finite Group:
 * $k \divides \order {\Z'_m}$

By the definition of the Euler $\phi$ function:
 * $\order {\Z'_m} = \map \phi m$

Thus: