Example 11 ([19.3]) Let $n>2$ be a given natural number, and let the set $V_{n}=\{k n+1: k=1,2, \cdots\}$. A number $m \in V_{n}$ is called an irreducible number in $V_{n}$ if there do not exist $p \in V_{n}, q \in V_{n}$ such that $m=p q$. Prove: There exists $r \in V_{n}$ that can be expressed as a product of irreducible numbers in $V_{n}$ in more than one way.

This problem shows that the uniqueness of representing a positive integer as a product of "primes" is not necessarily a valid property. In certain sets of integers, such a property does not hold. There are many such examples, and the appendix also provides such examples.