# 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 now it is true . they are rational numbers – math enthusiastic Jul 29 '17 at 5:25
• @John Wayland Bales I solved it but I don't know how to use match Jax and write the answer.can you help me? – math enthusiastic Jul 29 '17 at 17:37
• @John Wayland Bales thanks a lot I will try it – math enthusiastic Aug 1 '17 at 11:49