k-Integrality, an extension of total unimodularity

Let A be a matrix of integers, and b a vector of integers such that all elements of b have a common divisor k. When does the polyhedron P(A, b) = {x | x >= 0, Ax @? b} have integral extreme points only? We give a necessary and sufficient condition for this to happen and provide its special version for the case where k is a prime number. For k = 2, we give applications, including a new result for the obnoxious facility location problem.