Question on sufficient statistics

Question on sufficient statistics

Let ${\bf X}=(X_1,...,X_n)$ be a random sample from the pdf
$$f(x\mid\mu,\sigma)=\frac{1}{\sigma}e^{\frac{-(x-\mu)}{\sigma}}\;\; >
,\;\; \mu<x<\infty\;,\;0<\sigma<\infty.$$
Find a two-dimensional sufficient statistic for $(\mu,\sigma)$.
The Factorization Theorem says that I should be able to find $T_1({\bf
x})$ and $T_2({\bf x})$ such that we have the decomposition
$$\hat{f}({\bf x}\mid\mu,\sigma)=g(T_1({\bf x}),T_2({\bf
x})\mid\mu,\sigma)\cdot h({\bf x}).$$
Where $\hat{f}$ is the joint distribution for $\bf X$.
I just want to make sure I'm not missing something obvious, since by
independence I can write $\hat{f}$ as
$$\hat{f}({\bf
x}\mid\mu,\sigma)=\hat{f}(x_1,...,x_n\mid\mu,\sigma)=(\frac{e^{\frac{\mu}{\sigma}}}{\sigma})^ne^{\frac{-1}{\sigma}\sum_{k=1}^nx_k}\cdot
1.$$
It appears that just setting $T_1({\bf x})=\sum_{k=1}^nx_k$ is a statistic
sufficient to determine both $\mu$ and $\sigma$. Which worries me since
why would the book ask specifically for a two dimensional statistic if a
one dimensional one was all that was necessary to obtain sufficiency with
regard to both parameters.