Euler's Theorem (Number Theory)/Corollary 1

Corollary to Euler's Theorem (Number Theory)

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:

$a^{\paren {p - 1} p^{n - 1} } \equiv 1 \pmod {p^n}$

Proof

We have that Divisor Relation is Transitive.

Since $p \divides p^n$, it follows that:

$p^n \nmid a$

From Euler's Theorem (Number Theory):

$a^{\map \phi {p^n} } \equiv 1 \pmod {p^n}$

From Euler Phi Function of Prime Power:

$\map \phi {p^n} = \paren {p - 1} p^{n - 1}$

Then:

$a^{\paren {p - 1} p^{n - 1} } \equiv 1 \pmod {p^n}$

$\blacksquare$