If $f:\mathbb N\to\mathbb Z$ satisfies:
$$\forall n,m\in\mathbb N\,, n+m\mid f(n)+f(m)$$
How to show that this implies:
$$\forall n,m\in\mathbb N,\,n-m\mid f(n)-f(m)?$$
I was almost incidentally able to prove this by classifying such functions, but that seems circuitous for such a result. Is there a proof that is (more) direct?