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find all rational numbers $a,b \in \mathbb{R}$ such that $3\, | \, \left(a+b+\dfrac{1}{a}+\dfrac{1}{b}\right)$

I solved it in integer numbers but I couldn't solve in in rational numbers because I can't use to be divisible something with some thing else.

I also supposed $a=\dfrac{m}{n}$ such that $gcd(m,n)=1$ and $b=\dfrac{p}{q}$ such that $gcd(p,q)=1$ but I was not able to continue to a solution.

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    $\begingroup$ sorry now it is true . they are rational numbers $\endgroup$ – math enthusiastic Jul 29 '17 at 5:25
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    $\begingroup$ @John Wayland Bales I solved it but I don't know how to use match Jax and write the answer.can you help me? $\endgroup$ – math enthusiastic Jul 29 '17 at 17:37
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    $\begingroup$ @John Wayland Bales thanks a lot I will try it $\endgroup$ – math enthusiastic Aug 1 '17 at 11:49
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    $\begingroup$ @JohnWaylandBales I post the answer with photo's links, please edit the answer $\endgroup$ – math enthusiastic Aug 1 '17 at 13:08
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    $\begingroup$ @Χpẘ I post the answer with photoes links can you edit it and write the solution.please $\endgroup$ – math enthusiastic Aug 1 '17 at 14:04
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sorry I am not good at using math Jax I wrote a neat solution and I took photos if some body is able to use math Jax please edit the answer and write the solution the first link is a solution and the next is proof of a theorem that I used it in my solution

http://s8.picofile.com/file/8302280634/20170801_170848_1.jpg

http://s8.picofile.com/file/8302280926/20170801_170859_1.jpg

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