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

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?

• $aRb$ iff $5\mid a-b$ iff $5\mid b-a$. – Did Dec 4 '11 at 22:34
• a hint: $(a+4b) + (b+4a) = (5a + 5b)$ – deinst Dec 4 '11 at 22:36
• I've changed "Symmetric proof" to "Proof of Symmetry". "Symmetric proof" reads as if you are saying that the proof is (somehow) symmetric, rather than that the proof is a proof about symmetry. "Symmetry proof" would have been okay, too... – Arturo Magidin Dec 4 '11 at 23:28

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.
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 $a+4b$?
• @deinst's comment cleared up the symmetry part for me, but I don't see why adding $a+4x$ to $x+4b$ helps us solve the transitivity part of the proof? – David Chouinard Dec 4 '11 at 22:52
• $(a+4x)+(x+4b)=(a+4b)+5x$, so $a+4b=(a+4x)+(x+4b)-5x$. What do you know about each of the three terms on the righthand side? – Brian M. Scott Dec 4 '11 at 22:59