A generalization of a result of Sylvester's

Abstract Let v1,…,vn be vectors in Zn with D = det(v1,…,vn) > 0. Let vn + 1 be in the cone generated by v1,…,vn and such that v1,…,vv, vn + 1 generate Zn as a Z-module. There exists a unique “largest“ χ not expressible as a nonnegative integer combination of v1,…,vn, vn + 1 and χ = Dvn + 1 − (v1 + … vn + vn + 1).