Why is $a^n - b^n$ divisible by $a-b$? I did some mathematical induction problems on divisibility


*

*$9^n$ $-$ $2^n$ is divisible by 7.

*$4^n$ $-$ $1$ is divisible by 3.

*$9^n$ $-$ $4^n$ is divisible by 5.


Can these be generalized as
$a^n$ $-$ $b^n$$ = (a-b)N$, where N is an integer?
But why is $a^n$ $-$ $b^n$$ = (a-b)N$ ?
I also see that $6^n$ $- 5n + 4$ is divisible by $5$ which is $6-5+4$ and $7^n$$+3n + 8$ is divisible by $9$ which is $7+3+8=18=9\cdot2$.
Are they just a coincidence or is there a theory behind?
Is it about modular arithmetic?

It can be generalized as:
$$a^n-b^n = (a-b)(a^{n-1}+a^{n-2}b+\cdots +b^{n-1})$$
If you are interested in a modular arithmetic point of view, since $a \equiv b \pmod{a-b},$ $a^n \equiv b^n \pmod{a-b}.$
Your last two examples are true because what you are essentially doing is plugging in $n=1.$
