Is it true that exists a natural number (zero excluded) that is divisible by every natural number?

Question

I have the following formula:

$(\exists x)(\forall y) r(x,y)$

Is this formula true in a model where r(x,y) is a binary predicate interpreted as "x is divisible by y" and the universe is all natural numbers except for zero? Why?

Answer 1

Why are you excluding $0$?

The statement is true for $0$ (because $x*0= 0$ is true for all $x$).

But for any $y\ne 0$ the statement, which is equivalent to state $\frac xy$ is an integer, is obviously false. What if $y > x$? Then if $x>0$ we have $0< \frac xy < 1$. What if $y = 2x$. The we have $\frac x{2x} = \frac 12$.