find all rational numbers $a,b$ such that $3$ divides $a+b+\frac{1}{a}+\frac{1}{b}$ 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.

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
