For what values of $m \in N$, the following divisibility $x+y / x^m+y^m$ holds?

Value of $m \in N$ for which $x^m+y^m$ is divisible by $x+y$. Let $P(m)=x^m+y^m$ is divisible by $x+y$. Let us consider $m=1$. $P(1): x+y$ is divisible by $x+y ; \therefore P(1)$ is true $\Rightarrow m=2 \Rightarrow P(2): x^2+y^2$ is divisible by $x+y$ which not true $\Rightarrow P(2)$ is not true. $\Rightarrow m=3 \Rightarrow P(3)=x^3+y^3$ is divisible by $x+y$. $\therefore x^3+y^3=(x+y)\left(x^2-x y+y^2\right)$ $\therefore P(3)$ is true. Option 1 Even number wrong ( $m=2$ not valid). Option 2 odd number right. Option 3 All natural number wrong ( $\because m=2$ not valid) Option 4 Only when $m=1$ wrong because valid for $m=3$ it is possible.