Question 61
Let $\mathbb{N}$ denote the set of positive integers. Find all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ such that the following assertions are true for every pair of positive integers $(x, y)$:
- $x$ and $f(x)$ have the same number of positive divisors.
- If $x$ does not divide $y$ and $y$ does not divide $x$, then
$$
\operatorname{gcd}(f(x), f(y)) \gt f(\operatorname{gcd}(x, y))
$$
Here $\operatorname{gcd}(m, n)$ is the largest positive integer that divides both $m$ and $n$.
- $x$ and $f(x)$ have the same number of positive divisors.
- If $x$ does not divide $y$ and $y$ does not divide $x$, then
$$
\operatorname{gcd}(f(x), f(y)) \gt f(\operatorname{gcd}(x, y))
$$
Here $\operatorname{gcd}(m, n)$ is the largest positive integer that divides both $m$ and $n$.