Answer
See below.
Work Step by Step
1. Test for $n=1$, $a+b$ is a factor of $a^3+b^3$, it is true.
2. Assume the statement is true for $n=k$, that is $a+b$ is a factor of $a^{2k+1}+b^{2k+1}$,
3. For $n=k+1$, we have $a^{2k+3}+b^{2k+3}=a^2a^{2k+1}+b^2b^{2k+1}=a^2a^{2k+1}+a^2b^{2k+1}-a^2b^{2k+1}+b^2b^{2k+1}=a^2(a^{2k+1}+b^{2k+1})-b^{2k+1}(a^2-b^2)$ 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 $n$.
