How to prove these statements : $3 \mid (10^{n+1} + 10^n + 1)$ and $(a-b) \mid (a^n - b^n)$

Question

I was looking at some proof questions and had difficulty answering a few of them How do I prove these statements below:

1) $3 \mid (10^{n+1} + 10^n + 1)$

2) $(a-b) \mid (a^n - b^n)$

Answer 1

Assuming all variables are integers, and $n\geq0$, since otherwise the questions do not make much sense.

In the first question use $10\equiv1\pmod3$ and compute the right hand side modulo$~3$. In the second question use $a\equiv b\pmod{a-b}$ and compute the right hand side modulo$~a-b$.