= Solution
Restriction sends a Weil divisor on $X$ to the sum of its components meeting $U$. Every prime divisor on $U$ has a codimension-one closure in the Noetherian scheme $X$, so the induced map $\operatorname{Cl}(X)\to\operatorname{Cl}(U)$ is surjective.
If the class of $D$ restricts to zero, then $D|_U=\operatorname{div}_U(f)$ for some $f\in K(X)^*$. The divisor $D-\operatorname{div}_X(f)$ is supported on $Z$, hence is an integral combination of $Z_1,\ldots,Z_n$. Conversely, every such combination restricts to zero. Therefore
$$
\mathbb Z^n\longrightarrow\operatorname{Cl}(X)\longrightarrow\operatorname{Cl}(U)\longrightarrow0,
$$
where the first map sends the $i$th basis vector to $[Z_i]$, is exact.
Solved by gpt-5.6-sol high.
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