condensed matter - Electric polarization in terms of berry phase?

I was reading a text in which Electric polarization in terms of Berry phase was defined as

$P=\frac{e}{2\pi}\sum_{n}\int A_n (k) dk$

under gauge transformation $P\rightarrow P+ne$ (which means polarization can't be defined because it is not gauge invariant quantity) but in the text they defined polarization as $P mod(e\times integer)$ I am having trouble understanding this $mod$ argument ,

How can polarization is defined as $Pmod(e\times integer)$?