Let $f: X' \rightarrow X$ be the blowing up of a nonsingular variety $X$ along a nonsingular subvariety $Z$ of codimension $r \geq 2$, and let $Z' = f^{-1}(Z)$. Denote $\mathrm{Cl}(X')$ the divisor class group of $X'$. Why is the map $\mathbb{Z} \rightarrow \mathrm{Cl}(X')$, $q \mapsto q \cdot Z'$, injective? That is, why is not the divisor $q \cdot Z'$ principal for $q \neq 0$?
Thinking about this problem I rose the following question: let $(f, f^{\#}): X \rightarrow Y$ be a birational morphism of integral schemes and let $\xi$ be the generic point of $X$. Then $f_{\xi}^{\#}: K(Y) \rightarrow K(X)$ is an isomorphism between the function fields. Suppose that $f^{-1}(Z)$ is a prime divisor on $X$ for each prime divisor $Z$ on $Y$. Let $\mathrm{div}(g) = \sum n_{Z} \cdot Z$, $g \in K(Y)$, be a principal divisor. Is this true that $\mathrm{div}(f_{\xi}^{\#}(g)) = \sum n_{Z} \cdot f^{-1}(Z)$ Is this true in the case above?
Thanks.
P.S.: I do not know cohomology of sheaves, not yet.