Let a,b, and c be complex numbers such that:
$\frac{ab+1}{3b}=\frac{bc+1}{4c}=\frac{ca+1}{5a}=2020.$
My question is to find the remainder when $abc+\frac{1}{abc}$ is divided by $10^6$, so not to necessarily find what a, b, and c are. I have thought so far to multiply by the LCM $60abc$ so that I would have an abc on the right side of the equation, but I am not sure what to do from there.
Any suggestions are greatly appreciated! Thank you