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Problem 4. Let $\mathbb{N}$ be the set of positive integers. Find all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ that simultaneously satisfy the following properties:
(i) $f(mn) = f(m)f(n)$ for all positive integers $m$ and $n$;
(ii) There exists a positive integer $c$ such that $f(n) \le n^c$ for all positive integers $n$;
(iii) The numbers $f(n) + m$ and $f(m) + n + 1$ are coprime for all positive integers $m$ and $n$.
Solution 1Solution 2Solution 3Solution 4
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