Euler's Theorem

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

Let $\phi \left({m}\right)$ be the Euler $\phi$ function of $m$.

Then:
 * $a^{\phi \left({m}\right)} \equiv 1 \pmod m$

Proof
Let $\left[\!\left[{a}\right]\!\right]_m$ denote the residue class modulo $m$ of $a$.

Since $a \perp m$, it follows by Reduced Residue System under Multiplication forms Abelian Group that $\left[\!\left[{a}\right]\!\right]_m$ belongs to the abelian group $\left({\Z'_m, \times}\right)$.

Let $k = \left|{\left[\!\left[{a}\right]\!\right]_m}\right|$ where $\left|{\cdots}\right|$ denotes the order of a group element.

By Order of Element Divides Order of Finite Group:
 * $k \mathrel \backslash \left|{\Z'_m}\right|$

By the definition of the Euler $\phi$ function:
 * $\left|{\Z'_m}\right| = \phi \left({m}\right)$

Thus: