elementary number theory - Prove by induction that $a-b|a^n-b^n$

Given $a,b,n \in \mathbb N$, prove that $a-b|a^n-b^n$. I think about induction. The assertion is obviously true for $n=1$. If I assume that assertive is true for a given $k \in \mathbb N$, i.e.: $a-b|a^k-b^k$, I should be able to find that $a-b|a^{k+1}-b^{k+1}$, but I can't do it. Any help is welcome. Thanks!

Answer

To complete the induction, note that

$a^{k + 1} - b^{k + 1} = a^{k + 1} - a^kb + a^kb - b^{k + 1} = a^k(a - b) + b(a^k - b^k), \tag{1}$

then simply observe that

$(a - b) \mid a^k(a - b), \tag{2}$

which is obvious, and that

$(a - b) \mid (a^k -b^k) \tag{3}$

by the induction hypothesis

$(a - b) \mid (a^k - b^k). \tag{4}$

Since $a - b$ divides both summands, it divides their sum.QED

Hope this helps. Cheerio,

and as always,

Fiat Lux!!!