Are these valid proofs? Specifically, is the way that I introduced the integer k the right way to do that? Thanks.

Let R be the relation onZgiven by xRy iff x divides y.

Proof (Reflexive)

Suppose x is inZ. Consider integer k = 1. Then, x = (1)x = kx, and x divides x.

Thus, xRx and R is reflexive.

Proof (Symmetric)

Counterexample: Suppose (x, y) = (1, 10). Let k = 10. Then, y = 10 = (10)1 = kx, and xRy.

But (y, x) = (10, 1) and y = 1 = (k)10 for k = 1/10. Thus, y does not divide x

and (y, x) is not in R. So, R is not symmetric.

Proof (Transitive)

Suppose (x, y) and (y, z) are both in R. Then there exists integers k and n such that y = kx and z = ny.

So, z = ny = n(kx) = (nk)x. Since the product of integers is again an integer, nk is an integer.

Thus, x divides z and xRz. Therefore, R is transitive.