# Minimum sufficient statistic for logistic regression model

For the question in the link below, I am seeking the minimal sufficient statistic for $$\theta$$={$$\beta_1$$,$$\beta_2$$} in the linear regression model given.

I have taken the ratio of likelihoods $$L(x, \theta)$$/$$L(y,\theta)$$ which has given me the result below:

I know that I am looking for a condition that makes the likelihood ratio independent of $$\theta$$. From the first term in the product, when the sum of all $$x_i$$ is equal to the sum of all $$y_i$$, this term is made independent of $$\theta$$.

What about the second term in this product? Is there a second condition from this fraction that I need to synthesise into the solution, or have I already shown that the minimal sufficient statistic is the sum of all $$x_i$$? I can't find a way to algebraically break up that second fraction so that the product symbol goes away on each side.

Thanks.