I'm asked to show that $X_{1:n}$ (the minimum order statistic) is sufficient for $\eta$, in the case of a random sample $(X_1, ... , X_n)$ where $X_i\sim EXP(1,\eta$) (this is the two-parameter exponential distribution $EXP(\theta,\eta):$ $(1/\theta)\exp(-(x-\eta)/\theta)$, $x>\eta$; in this case $\theta=1$), by using the "factorization method", that is, writing $f(x_1,...,x_n;\eta)$ as $g(s,\eta)h(x_1,...,x_n)$, where $S$ is the statistic ($X_{1:n}$, in this case), $g(s;\eta)$ does not depend on $x_1,...,x_n$ except through $s$, and $h(x_1,...,x_n)$ does not involve $\theta$.
I have $f(x_1,...,x_n;\eta)=\exp(n\eta)\exp(-\sum_{i=1}^n x_i)$. The sum in the exponential is the same as $\sum_{i=1}^n x_{i:n}$, but I need an expression that involves $x_{1:n}$ in one factor and the $x_i$'s in the other. I don't know how to do this.
I know there are formulas for joint pdf's of any set of order statistics (quite lengthy), but I really don't know how to proceed.
Thank you.