Factorization of $x^b+1$

Question

I came across the result that $(x^a+1)|(x^b+1)$ if and only if $\frac{b}{a}$ is odd. Any intuitive reason why, though? What about $\frac{b}{a}$ being odd makes this true?

Answer 1

This follows from the fact that, if $k$ is odd,$$x^k+1=(x+1)\left(x^{k-1}-x^{k-2}+\cdots-x+1\right).$$