Find all surjective functions
![f: \mathbb{N} \mapsto \mathbb{N}](/media/m/0/6/4/064f5e79a7f8bd81b5e7163e70e87d6b.png)
such that for every
![m,n \in \mathbb{N}](/media/m/c/4/2/c428fc23c2ec4092eae3dee25f8bd1bf.png)
and every prime
![p,](/media/m/1/3/5/1359ba7e6926698785a400ad4053aed1.png)
the number
![f(m + n)](/media/m/b/2/c/b2c868cd705dd0ced359d370b8170bda.png)
is divisible by
![p](/media/m/1/c/8/1c85c88d10b11745150467bf9935f7de.png)
if and only if
![f(m) + f(n)](/media/m/4/c/d/4cd65d6ae391b2476e07bde1f85b8212.png)
is divisible by
Author: Mohsen Jamaali and Nima Ahmadi Pour Anari, Iran
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Find all surjective functions $f: \mathbb{N} \mapsto \mathbb{N}$ such that for every $m,n \in \mathbb{N}$ and every prime $p,$ the number $f(m + n)$ is divisible by $p$ if and only if $f(m) + f(n)$ is divisible by $p.$
Author: Mohsen Jamaali and Nima Ahmadi Pour Anari, Iran