Is this relation R transitive?

Question

There's a problem on my homework that asks if a "binary relation is transitive" and I'm really confused on how I should start this problem.

Most examples of proving transitivity online are of sets, but this question asks about the relation $R: R(x,y) = ∃k.9x + 15y = 21k$.

I know that the definition of transitivity is $∀x,y,z.R(x,y) ∧ R(y,z) → R(x, z)$. Can I use the variable $k$ as $z$?

Answer 1

The problem is well posed only if we look for $k \in \mathbb{Z}$

Counterexample:

For $x=4, y=-1$ we have: $R(x,y): 9*4+15*(-1)=21*1$, so $k_1=1$

For $y=-1, z=2$ we have: $R(y,z):9*(-1)+15*2=21*1$, so $k_2=1$

But $R(y,z)$ implies that $\exists k_3:$

$9*4+15*2=66=21k_3$ which is simply not true.