P-adic Valuation of Difference of Powers with Coprime Exponent/Proof 1

Proof
We have:
 * $x^n - y^n = \paren {x - y} \paren {x^{n - 1} + \cdots + y^{n - 1} }$

We have to show that:
 * $p \nmid x^{n - 1} + \cdots + y^{n - 1}$

Because $x \equiv y \pmod p$:
 * $x^{n - 1} + \cdots + y^{n - 1} \equiv x^{n - 1} + x^{n - 1} + \cdots + x^{n - 1} = n x^{n - 1} \pmod p$

Because $p \nmid x$ and $p \nmid n$:
 * $p \nmid n x^{n - 1}$