Consider a random sample of size $n$ froma gamma distribution, $X_i\sim GAM(\theta, \kappa)$, with $\kappa$ being the shape parameter and $\theta$ being the scale parameter and let $\bar X=\dfrac{1}{n}\sum X_i$ and $\tilde X=(\prod X_i)^{1/n}$ be the sample mean and geometric mean, respectively.
Show that the conditional distribution of $\bar X |_{\tilde X = \tilde x}$ does not depend on $\kappa$.
I have absolutely no idea how to find this conditional distribution. Even if there are some strategies to go about solving this problem without solving for the conditional distribution explicitly, I don't know what they are. How would you show this? Is there a general strategy for finding conditional distributions of statistics that I don't know about? I'm so lost.