how to prove independence in this case

Question

The question is :

$X_1,...,X_n$ are i.i.d.$Uniform(0,\theta)$. Let $X_{(n)}$ denote the maximum of these $n$ random variables. Prove that $\frac{X_1}{X_{(n)}}$ and $X_{(n)}$ are independent.

What I have now is that I can prove $X_{(n)}$ is the minimal sufficient statistic, but I do not know how to relate this to the proof of independence. Any suggestions?

Answer 1

Oh the answer is that , $\frac{X_1}{X_{(n)}}$ is an ancillary sufficient statistic, and we also can prove that $X_{(n)}$ is complete sufficient statistic. So by Basu's Lemma we conclude independence.