2. Find all functions $f: \mathbb{N} \rightarrow \mathbb{Z}$ such that

$$
f(a)+f(b)+f(c)
$$

is divisible by $a+b+c$ for any natural (not necessarily distinct) $a, b, c$.
2. Substitute $(a, b, c)=(1, b+1, c)$ and $(a, b, c)=(2, b, c)$ and subtract. We get that $f(1)+f(b+1)-f(2)-f(b)$ is divisible by $b+c+2$. Due to the arbitrariness of $c$, we obtain $f(1)+f(b+1)-f(2)-f(b)=0$, so $f(b+1)-f(b)=f(2)-f(1)=$ const, hence $f(x)=A x+B$ with constant integers $A, B$. It is easy to see that only $B=0$ works.