Question 27
Find all functions $f: \mathbb{N} \to \mathbb{N}$ such that:
* for any positive integers $a$ and $b$ which are not relatively prime we have $f(a)f(b) = f(ab)$,
* for any positive integers $a$ and $b$ there exists a non-degenerate triangle with the sides of lengths $f(a)$, $f(b)$ and $f(a + b - 1)$.
(Note: A triangle is non-degenerate if its three vertices do not lie on a common line.)
* for any positive integers $a$ and $b$ which are not relatively prime we have $f(a)f(b) = f(ab)$,
* for any positive integers $a$ and $b$ there exists a non-degenerate triangle with the sides of lengths $f(a)$, $f(b)$ and $f(a + b - 1)$.
(Note: A triangle is non-degenerate if its three vertices do not lie on a common line.)