Asian Online Journals (www.ajouronline.com) 74 New Constructive Approach to Solve Problems of Integers' Divisibility

ABSTRACT--- This paper aims at introducing a new constructive approach to solve problems in elementary number theory. It starts with a comprehensive analysis on present approaches to solve problems related with divisible features of consecutive integers, which include consecutive positive integers, consecutive positive odd integers and consecutive positive even integers; then it detailly demonstrates advantages and disadvantages of the present-applied approaches in their deducing process, especially the conflicts in proving the almost same-stated statements; in the end the paper puts forward a new constructive approach and uses it to have a new proof for the three fundamental theorems: for any positive integer n and among n consecutive positive integers there exists one and only one that can be divisible by n; for any positive odd integer p and among p consecutive positive odd integers there exists one and only one that can be divisible by p; for a positive even integer w and among w consecutive positive even integers, there exist exactly two that can be divisible by w. The new constructive proof is valuable for more extensive utilities in elementary number theory. Keywords---- Mathematical proof;Elementary number theory;Divisibility; Consecutive Integers;Residue system