Prove that any finite poset is isomorphic to a subset of $\mathbb {N}$ ordered by divisibility rule

Question

Prove that any finite poset is isomorphic to a subset of $\mathbb {N}$ ordered by divisibility rule.

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I have no idea how to prove this so any hints would be much appriciated.

Answer 1

Given any finite poset $\{x_1,x_2,...,x_n\}$ and any set of $n$ distinct primes $\{p_1,p_2,...,p_n\}$ (e.g. the first $n$ primes), the poset can be embedded in $(\mathbb{N}, \vert)$ via the map that sends $x_i$ to the product of the primes $p_j$ over those indices $j$ for which $x_j \le x_i$.