Let $a$ and $b$ be integers with $a-b\neq0$ and $n$ a positive integer . Show that

Question

Let $a$ and $b$ be integers with $a-b\neq0$ and $n$ a positive integer . Show that $\left(\dfrac{a^{n}-b^{n}}{a-b},\ a-b\right)=(n(a,b)^{n-1},a-b)$.

Note that $(x,y)=\gcd(x,y)$. Using the factorization $a^{n}-b^{n}=(a-b)(a^{n-1}+\dots+b^{n-1})$, I basically called one side $d$ and the other $f$ and tried to show that $d\mid f$ and $f\mid d$.