TITLE: Multiplicative structure without unique prime factorisation
QUESTION [7 upvotes]: The subset $N:=\{3n+1\colon n\in\mathbb{N}\}$ is closed under multiplication. 4, 10 and 25 are prime numbers in $N$. We have $100=4\cdot 25=10\cdot 10$, hence factorisation with prime numbers in $N$ is not unique.
Is there a subset $M$ of $\mathbb{N}$ that is closed under multiplication such that there is a number $n\in M$ with $n=\prod_{i=1}^jp_i=\prod_{i=1}^kq_i$ where $j<k$ and the $p_i$'s and $q_i$'s are prime numbers in $M$?
If there is no such subset in $\mathbb{N}$, is there another example where the number of prime elements in a factorisation is not unique?

REPLY [2 votes]: Take $M = \{5n+1 : n \in \mathbb{N}\}$. Then the number $12996$ has two prime factorizations, one into $2$ and one into $3$ primes: $$ 12996 = 76 \times 171 = 6 \times 6 \times 361.$$