Relations involving division

Question

Can someone explain me how to do it?
Let R be a relation on integers such that xRy and iff 3|5x+7y.
Show that relation is reflexive ( I am done with it!) and symmetric (I need help with this one).

Answer 1

I'll leave it to you to justify this carefully:

$3|(5x+7y) \iff 3|(2x+y) \iff 3|2(2x+y) \iff 3|(4x+2y) \iff 3|(x+2y)\iff 3|(7x+5y)$

Answer 2

Here is another way, similar to Casteels' (so I copy his notation) but more clear in some ways:

$$xRy \iff 3|(5x+7y) \iff 3|(-x+y) \iff 3|(x-y)$$ $$\iff 3|(7x+5y) \iff 3|(5y+7x) \iff yRx$$

And here is yet another way, even closer to Casteels'.

$$xRy \iff 3|(5x+7y) \iff 3|2(5x+7y) \iff 3|(10x+14y)$$ $$\iff 3|(7x+5y) \iff 3|(5y+7x) \iff yRx$$