Define the function $\displaystyle f: \mathbb{N} \longrightarrow \mathbb{N}$ recursively by: $\displaystyle \sum_{d \mid n}f(d)=2^n, \ \forall n \in \mathbb{N}.$ Prove that $\displaystyle f(n)$ is divisible by $\displaystyle n$ for all $\displaystyle n \in \mathbb{N}.$