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

Proof
We have $x^n-y^n=(x-y)(x^{n-1}+\cdots+y^{n-1})$.

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

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

Because $p\nmid x$ and $p\nmid n$, $p\nmid nx^{n-1}$.