Euler's Theorem/Corollary 1

Corollary to Euler's Theorem
Let $p^n$ be a prime power for some prime number $p > 1$.

Let $a$ be an integer not divisible by $p: p \nmid a$.

Then:
 * $a^{\left({p - 1}\right) p^{n - 1}} \equiv 1 \pmod {p^n}$

Proof
We have that Divisor Relation is Transitive.

Since $p \divides p^n$, it follows that $p^n \nmid a$.

From Euler's Theorem:
 * $a^{\phi \left({p^n}\right)} \equiv 1 \pmod {p^n}$

From Euler Phi Function of Prime Power:
 * $\phi \left({p^n}\right) = \left({p - 1}\right) p^{n - 1}$

Then:
 * $a^{\left({p - 1}\right) p^{n - 1}} \equiv 1 \pmod {p^n}$