Show that $R = \{ (x,y) \in \mathbb{Z} \times \mathbb{Z} : 4 \mid(5x+3y)\}$ is an equivalence relation. 
Let $R$ be a relation on $\mathbb{Z}$ defined by $$ R = \{ (x,y) \in \mathbb{Z} \times \mathbb{Z}  :  4 \mid (5x+3y)\}.$$ Show that $R$ is an equivalence relation. 

I'm having a bit of trouble with this exercise in my book and I am trying to study. Can anyone give guidance for this? I know we have to show reflexivity, symmetry, and transitivity, but I don't think what I have on my paper is completely right. I would appreciate other people's opinions on what the solution should be.

In general, let $a$ and $b$ be integers and $m$ a positive integer.  Then, $$R:=\big\{(x,y)\in\mathbb{Z}\times\mathbb{Z}\,|\,ax+by\text{ is divisible by }m\big\}$$ is an equivalence relation on $\mathbb{Z}$ if and only if $m$ divides $a+b$.  Furthermore, if $m$ divides $a+b$, then the set of equivalence classes of $\mathbb{Z}$ under $R$ is precisely $\mathbb{Z}/R = \mathbb{Z}/d\mathbb{Z}$ (i.e., $R$ is exactly congruence modulo $d$), where $d:=\frac{m}{\gcd(m,a)}=\frac{m}{\gcd(m,b)}$.
