Prove $13|19^n-6^n$ by congruences I am trying to prove  $13|19^n-6^n$. With induction its not so bad but by congruences its quite difficult to know how to get started.
Any hints?

Because $x^n - y^n$ is divisible by $x-y$ as
$$x^n - y^n = (x-y)\sum_{i=0}^{n-1}x^iy^{n-1-i}$$
Substitute $x=19$ and $y = 6$.

$$\begin{align*}
x^n-y^n =& x^n\left[1-\left(\frac yx\right)^n\right]\\
=& x^n \left[1+\left(\frac yx\right)+\left(\frac yx\right)^2+\cdots+\left(\frac yx\right)^{n-1}\right]\left[1-\left(\frac yx\right)\right]\\
=& \left(x^{n-1}y^0 + x^{n-2}y + x^{n-3}y^2 + \cdots + x^0y^{n-1}\right) (x-y)
\end{align*}$$
