In this paper，we have given a concept of greatest common factor and least common multiple in the nonempty set of integers with the nature of usual greatest common factor and least common multiple. Applying this concept，we incorpoate the characteristic number of a group's element and the characteristic numbers of a ring and field. Besides，We have proved that the greatest common factor and the least common muliple are existential for every nonempty set of integers and，unique for definite nonempty set of integers.