Consider a variety $C$ which is a complete intersection of $X$ and $Y$ in $\mathbb{P}^n$ such that the degrees of $X$ and $Y$ are coprime, and so the degree of $C$ is the product of the degrees. Consider now a case where $Y$ is an hypersurface in $\mathbb{P}^n$.
If we take an effective divisor $D'$ in $X$, whose degree is $k$ and that not contains $X$, I think we could say that the degree of $D=D'\cap Y$ is $k y$, where $y$ is the degree of $Y$.
Consider instead $D$ an effective divisor in $C$ whose degree can be written as $ky$, where $y$ is the degree of $Y$. Is it true that there is an effective divisor $D'$ in $X$ whose degree is $k$, and such that $D=D'\cap Y$?
Or can we add some hypotheses in order to make this true?