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Let $\mathbb N$ be the set of all positive integers. How can one find all functions $f: \mathbb N \to \mathbb N$ such that $$ k \mid f(m+n) \iff k \mid f(m)+f(n)$$ For all positive integers $k$.

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up vote 5 down vote accepted

Let $k,x,y$ be integers. Note that if $k | x \Longleftrightarrow k | y$ occurs for all $k$, then $x = y$. (If $x \neq y$, you could find some number dividing one but not the other). Therefore, you simply need to find all functions $f : \mathbb{N} \to \mathbb{N}$ such that $f(x+y) = f(x) + f(y)$. It turns out that the only such equations are of the form $f(x) = a x$, for some constant $a$ (See Cauchy's Functional Equation).

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