Show that the product of the sample observations is a sufficient statistic for theta if the random sample is taken from a sample with parameters alpha = theta and beta = 6.

So I need to make sure that $\displaystyle Y = \prod X_i$ is a sufficient estimator for theta, which is true if:

$\displaystyle \frac{f(x_1; \theta)...f(x_n; \theta)}{Y = \prod X_i}$ does not depend on theta.

But I keep getting getting 1 for the ratio. Am I correct?