How many $triplets?$

Solution:

From $xy = z -> x = z/y$

$xz = y -> x = y/z$

$x = yz$

$\therefore$ We can conclude that $x = z/y = y/z = yz (1)$

So, $z/y = y/z -> z = {+/- y}$

From $x = z/y -> x = {+/-1}$

From $(1)$ , we can see that $y/z = yz -> (yz^2 - y) = 0 -> y = 0 , z = {+/- 1}$

Since $x = {+/- 1} and z = {+/- 1}$ , from the equation, xz = y, we can create $4$ values of y resulting to have a $4 triplets$ . Also remember that $y = 0$ so, therefore, there are $5$ values of y resulting to have a $\boxed{5 triplets}$ .