Answer
See below.
Work Step by Step
1. Test for $n=1$, $a-b$ is a factor of $a^1-b^1$, it is true.
2. Assume the statement is true for $n=k$, that is $a-b$ is a factor of $a^k-b^k$,
3. For $n=k+1$, use the given hint, we have $a^{k+1}-b^{k+1}=a(a^k-b^k)+b^k(a-b)$ which contains two part with each part containing a factor of $(a-b)$, thus it is true for $n=k+1$,
4. We can conclude that the statement is true for any integer $n$.