I have the following statement to determine the truth value of: $$\exists x \forall y P(x, y),$$
$P(x, y)$ is the statement "$x$ divides $y$", meaning that meaning that $y = kx$ for some integer $k$. $x$ and $y$ are both positive integers.
My first issue here is my understanding of $\exists x \forall y$, which I believe translates to:
"There is some $x$ for every $y$ such that $x$ divides $y$".
Is this correct, and if so, would this result in a positive truth value as $1$ divides all values of $y$?
As an extension to this, how is best to summise my reasoning for this positive* truth value?
*or vice-versa for a false statement