5 Find the smallest positive integer $c$, such that the indeterminate equation $x y^{2}-y^{2}-x+y=c$ has exactly three sets of positive integer solutions.

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5. Factorizing the left side of the equation, we get
$$(y-1)(x y+x-y)=c$$

Notice that, for any positive integer $c$, there is a solution
$$(x, y)=(1, c+1)$$

When $c$ is a prime number, there is at most one additional set of positive integer solutions. Therefore, to make the equation have exactly 3 sets of positive integer solutions, $c$ should be a composite number.

By direct calculation, it is known that the smallest $c$ that has exactly 3 sets of positive integer solutions is 10, and they are
$$(x, y)=(4,2),(2,3),(1,11)$$

The smallest positive integer
$$c=10$$