$\displaystyle a,b,c\in\mathbb{Z}$ $\displaystyle a\sim b$ iff. $\displaystyle 2|(a+b)$

This is suppose to be an equivalence relation; however, when I did the transitivity, it didn't work out.

reflexive: $\displaystyle a\sim a$

$\displaystyle 2|(a+a)\rightarrow 2p=2a$ Yes reflexive.

symmetric: $\displaystyle a\sim b$ then $\displaystyle b\sim a$

$\displaystyle 2|(a+b) \rightarrow 2n=a+b\rightarrow 2n=b+a\rightarrow 2n|(b+a)$

Thus, $\displaystyle b\sim a$

transitive:

$\displaystyle a\sim b$, $\displaystyle b\sim c$, then $\displaystyle a\sim c$

$\displaystyle a\sim b\rightarrow 2m=a+b$

$\displaystyle b\sim c\rightarrow 2r=b+c$

Subtracted the equations:

$\displaystyle 2(m-r)=a-c$

Again, I am supposed to conclude transitive what is going wrong here?