Intersection of a divisor with a principal divisor

Question

I am new in the study of surfaces in the algebraic geometry point of view. I am studying chapter V of Hartshorne. At some point while studying I came accross with the following thought which I was wondering if its true since I can neither prove nor disprove it. Let $X$ be a smooth projective surface. I've read that the bilinear pairing $DivX\times DivX\to \mathbb{Z}$ respects linear equivalence and hence gives a symmetric bilinear map $Pic X\times Pic X\to \mathbb{Z}$. Does that in particular imply that for any divisor D and any principal divisor $(f)$ we have that $D\cdot (f)=0$. Does the last statement hold in general ? If so could I get some proof ?