Infinite set of standard primes as the set of standard prime divisors of a nonstandard number

Question

Suppose $(N, +, \cdot, 0, 1, <, =)$ is a proper elementary substructure of $(N^*, +^*, \cdot^*, 0^*, 1^*, =^*, <^*)$. Show that there exists some (infinite) $b$, where $b ∈ N^*$, such that for each prime number $p ∈ N$, $N^* \models p | b$ iff $p ∈ S$, where $S$ is some set containing infinitely many primes.

Answer 1

If we choose $S$ to be the collection of all the standard primes, then it suffices to set $b=H!$ for some infinite integer $H$. Since the standard model satisfies the elementary formula expressing the factorial, the same formula evaluated at $H$ will give a non-standard integer divisible by all the primes up to $H$ by elementary equivalence, and in particular by all the finite primes in $S$.