Let E be the Grassmann algebra of an infinite-dimensional vector space L over a field of characteristic zero. In this paper, we study the -gradings on E having the form , in which each element of a basis of L has -degree , or . We provide a criterion for the support of this structure to coincide with a subgroup of the group , and we describe the graded identities for the corresponding gradings. We strongly use Elementary Number Theory as a tool, providing an interesting connection between this classical part of Mathematics, and PI Theory. Our results are generalizations of the approach presented in Brandão A, Fidelis C, Guimarães A. -gradings of full support on the Grassmann algebra. J Algebra. 2022;601:332-353. DOI:10.1016/j.jalgebra.2022.03.014. See also in arXiv preprint, arXiv:2009.01870v1, 2020.