$5 \cdot 36$ positive integer $n>1, A_{n}=\{x \in \mathbb{N} \mid (x, n) \neq 1\}$. If for $x, y \in$ $A_{n}$, there is $x+y \in A_{n}$, then $n$ is called interesting. Find all interesting positive integers $n$ greater than 1.
(Bulgarian Mathematical Olympiad, 1994)
[Solution] Let $p$ be a prime number. We first prove that for all $n=p^{s} (s \in \mathbb{N})$ are interesting positive integers.

If $n=p^{s}$, and if there exists $x \in \mathbb{N}, (x, n) \neq 1$, then $p$ must be a factor of $x$. Therefore, for any $x, y \in A_{p^{s}}, x, y$ are multiples of $p$, so $x+y$ is also a multiple of $p$, thus $\left(x+y, p^{s}\right) \neq 1$, which means $x+y \in A_{p^{s}}$. Therefore, $n=p^{s}$ is an interesting positive integer greater than 1.

If the positive integer $n$ has at least two different prime factors, let $n=p^{s} q$, where $p$ is a prime number, $s$ is a positive integer, $q$ is a positive integer coprime with $p$, and $q>1$.

Since $\left(p, p^{s} q\right)=p, \left(q, p^{s} q\right)=q$, then $p \in A_{p^{s} q}, q \in A_{p^{s} q}$. If $(p+q, p^{s} q)=r, r>1, r \in \mathbb{N}$, since $p+q$ is a multiple of $r$, and $p$ and $q$ are coprime, then the positive integer $r$ greater than 1 is coprime with both $p$ and $q$, which contradicts the fact that $p^{s} q$ is a multiple of $r$. Therefore, $\left(p+q, p^{s} q\right)=1$, so $p+q$ is not in $A_{p^{s} q}$, thus $n=p^{s} q$ is not an interesting positive integer.

Therefore, all interesting positive integers greater than 1 are $n=p^{s}$, where $p$ is any prime number, and $s$ is any positive integer.