MEMO 2016 I4

Find all $f : \mathbb{N} \to \mathbb{N}$ such that $f(a) + f(b)$ divides $2(a + b - 1)$ for all $a, b \in \mathbb{N}$ .

Remark: $\mathbb{N} = \{ 1, 2, 3, \ldots \}$ denotes the set of the positive integers.