Proof that the relation $5 \mid (a + 4b)$ is symmetric and transitive

Question

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:

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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 $5 \mid (a + 4b)$.

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I've fiddled around with sample values, but I still don't see it. I'm pretty lost here. Thoughts?

Answer 1

Hint $\rm\,\ a\:R\:b \iff a-b\in 5\,\mathbb Z.\:$ If so, negating yields that $\rm\: b-a\in 5\,\mathbb Z\: $ hence $\rm\:R\:$ is symmetric. Transitivity follows by addition: $\rm \ a-b\:,\ b-c\:\in 5\:\mathbb Z\ \Rightarrow\ a-b + b-c\ =\ a-c \in 5\,\mathbb Z.$

Hence symmetry arises from $\rm\:5\,\mathbb Z\:$ being closed under negation, and transitivity arises from $\rm\:5\,\mathbb Z\:$ being closed under addition, i.e. from it being an additive subgroup of $\rm\mathbb Z.$ The innate algebraic structure will be brought to the fore when you study congruences and ideals of rings.

Answer 2

Symmetry: If $5\mid a+4b$, does $5\mid \big((a+4b)-(5a+5b)\big)$?

Transitivity: If $5\mid a+4x$ and $5\mid x+4b$, what can you say about $(a+4x)+(x+4b)$? How is this number related to $a+4b$?