Existence, Uniqueness and Counterexamples · Medium problem

There is a $y$ such that for every $x$, $x$ divides $y$.

For every $x$ there is a $y \ne x$ such that $x$ divides $y$.

For every $x$ there is a $y$ with $1 < y < x$ such that $y$ divides $x$.

There is an $x$ such that for every $y$, $x$ divides $y$.