Existence of inaccessible natural number divisible by every standard natural number under PA

Question

Let $P$ be the proposition that there exists a non-zero number that is divisible by every standard natural number.

Let $N$ be a non-standard model of PA.

Must $P$ be true in $N$?

Answer 1

If $N$ has an infinite number $h$, then the factorial $h !$ is divisible by every standard integer.

Answer 2

As an alternate proof, we can appeal to overspill. Let $\phi(x)$ be the statement $\exists y (\forall n < x (n | y))$. If $N$ is a model of PA, then for each standard $n$, $N \models \phi(n)$. By overspill, there must be some nonstandard $c$ such that $N \models \phi(c)$.