Elementary Number Theory: Divisibility proof

Question

Let $k,m,n \in N\setminus \{0\}$, s.t. $n=k\cdot m$. Show that $k$ is odd $\Rightarrow ∀ a,b \in Z: (a^m+b^m) \mid (a^n+b^n)$

In the first part of the task, I have already shown that $∀ a,b \in Z: (a^m-b^m) \mid (a^n-b^n)$

Answer 1

$a^{km} +b^{km} =(a^m +b^m )( a^{(k-1)m} -a^{(k-2)m} b^m +...+a^{m} b^{(k-2)m} +b^{(k-1)m} )$

Answer 2

Hint: if $k$ is odd then $$a^n + b^n = a^{km} + b^{km} = (a^m)^k - (-b^m)^k $$

Answer 3

You know that, if $k$ is odd, then $u+v \mid u^k + v^k$. Now let $u = a^m$ and $v = b^m$.