What does it meant :$4$ and $2(1+\sqrt{5}))$ without Great commone divisor?

Question

I have read in the book of counter example in Analysis by Bernard R. Gelbaum John M. H. Olmsted ,(page $18$) that $4$ and $2(1+\sqrt{5})$ without Great commone divisor however it's trivial they have no great commone divisor since $2(1+\sqrt{5})$ is not integer, I want some justification for this in the sense why these number were considered to look for them $\gcd$?