Existence of Number to Power of Prime Minus 1 less 1 divisible by Prime Squared/Examples/5

Example of Existence of Number to Power of Prime Minus 1 less 1 divisible by Prime Squared
The smallest positive integer $n$ greater than $1$ such that:
 * $n^{5 - 1} \equiv 1 \pmod {5^2}$

is $7$.

Proof
Only positive integers coprime to $5$ need be checked.

We have that $5^2 = 25$.

Thus: