Question 1027829
1. S is reflexive:  It is obvious that xSx is true.
2. S is symmetric:  xSy true implies that ySx is also true, because xy = ys is a perfect square.
3. S is symmetric.
xSy ==> xy = {{{k^2}}} for some integer k ==> {{{x = k^2/y}}} is an integer.
ySz ==> yz = {{{l^2}}} for some integer l ==> {{{z = l^2/y}}} is an integer.

==> xz is an integer that is a perfect square, since, {{{xz = (k^2/y)(l^2/y) = ((kl)/y)^2}}} 

Therefore S is an equivalence relation.

E.g., [16] = {(16,1), (8,2), (4,4), (2,8), (1,16)}