Positive integers $(x,y)$ satisfying $x^2-y^2= 12345678$

Question

I saw this question in one of the previous year Olympiad questions.

Let $S$ be the set of all ordered pairs of positive integers $(x,y)$ satisfying the condition : $x^2-y^2= 12345678$

Then what can be said about the number of elements in set $S$ ?

The solution of this question doesn't seem to be correct. Look at it :

Can someone provide me a reasonable approach ?

Answer 1

$x^2-y^2\equiv 12345678\equiv 2$ modulo $4$ which is not possible since squares are $0$ or $1$ modulo $4$. So the answer given is correct.