Show that the smallest factor greater than 1 of $p^n$ is the prime $p$.
Since $p$ is prime, its only factors are 1 and $p$.
The factors of
$$ p^n = \underbrace{p \times p \times \ldots \ p}_{n\text{ repetitions}}$$
are $1, p, p^2, p^3, \ldots, p^n$.
Note that $1 < p < p^2 < \ldots p^n$, since all primes are positive and strictly greater than 1.
So, the smallest factor, greater than 1, is $p$.