5. Find the number of integers $n$ such that the equation
$$
x y^{2}+y^{2}-x-y=n
$$
has an infinite number of integer solutions $(x, y)$.
(A) 0 ;
(B) 1 ;
(C) 2 ;
(D) 3 ;
(E) More than 3.
5. Answer: (C).
Rewrite the equation as
$$
x y^{2}+y^{2}-x-y=(y-1)(x(y+1)+y)=n .
$$

If $n=0$, then there are infinitely many integer solutions. Suppose $n \neq 0$, and the equation has infinitely many integer solutions. Then there exists a divisor $k$ of $n$ such that $y-1=k$ and $x(y+1)+y=n / k$ for infinitely many $x$. It forces $y+1=0$, i.e., $y=-1$. Then $n=2$.