Let $n$ be an odd integer and let $k$ be the number of elements in $U_n.$ What is the order of $U_{2n}$?
I have said $\left\lvert U_{2n}\right\rvert=\varphi(2n)=\varphi(2)\varphi(n)=\varphi(n)=k.$ (Since $\operatorname{hcf}(2,n)=1$).
Could someone verify whether this is correct or not?