Take the relation $R$ to be defined on the set of integers:
$$aRb \iff 5 \mid (a + 4b)$$
As part of a larger proof, I have to show that $R$ is both symmetric and transitive. I'm lost.
I see the first steps, but I can't find how to progress further. Here's what I have at this point:
Proof of Symmetry
We have to prove that if $5 \mid (a + 4b)$, then $5 \mid (b + 4a)$. Clearly, this is true if $a = b$, but apart from that, it's unclear in my mind.
Proof of Transitivity
We have to prove that if $5 \mid (a + 4x)$ and $5 \mid (x + 4b)$, then $5 \mid (a + 4b)$.
I've fiddled around with sample values, but I still don't see it. I'm pretty lost here. Thoughts?