i do understand the universe is set of all integers,

what you are doing wrong here is you are finding one y for all x which is wrong, you have to find at least one y for each and every x. there is a difference between

∀x ∃y P(x,y) and ∃y ∀x P(x,y)

The reason why S1 is false because for x=0 we can't find at least one y if the question would have been ∀x ∃y P(x,y) where x is not 0. Then it would be true.

yes i got u i was making a mistake i was taking a different path which was wrong and invalid. it means that all these values of x must divide some y or atleast one y..