TY - JOUR AU - Wang, Xingbo PY - 2014/06/15 Y2 - 2024/03/28 TI - New Constructive Approach To Solve Problems of Integers' Divisibility JF - Asian Journal of Fuzzy and Applied Mathematics JA - AJFAM VL - 2 IS - 3 SE - Articles DO - UR - https://ajouronline.com/index.php/AJFAM/article/view/1331 SP - AB - <p>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 <em>n</em> and among <em>n</em> consecutive positive integers there exists one and only one that can be divisible by <em>n</em>; for any positive odd integer <em>p</em> and among <em>p</em> consecutive positive odd integers there exists one and only one that can be divisible by <em>p</em>; for a positive even integer <em>w</em> and among <em>w</em> consecutive positive even integers, there exist exactly two that can be divisible by <em>w</em>. The new constructive proof is valuable for more extensive utilities in elementary number theory.</p> ER -