Why is this not a equivalence relation in the $\mathbb{Z}$ set?

Question

The relation $xRy$ on the set $\mathbb{Z}$ if $x + y$ is divisible by $3$.

$xRx = x + x = 2x$. In order to $2x$ be divisible by $3$, $x$ itself need to be divisible by $3$, it is not a problem is it?

$xRy = yRx$

$xRy = yRz = xRz$

I think the last two are immediate, so probably my error is in the first.

Answer 1

Note that $1R2$ ($1+2$ is divisible by $3$) and $2R4$ ($2+4$ is divisible by $3$) do not imply $1R4$ ($1+4$ is not divisible by $3$). Hence transitive property does not hold.

Also reflexive property $xRx$ is not satisfied when $x\equiv 1\pmod{3}$ or $x\equiv 2\pmod{3}$.

Answer 2

$xRx$ is not true for $x=1$. Therefore, the relation is not reflexive.