# New Constructive Approach To Solve Problems of Integers' Divisibility

## Keywords:

Mathematical proof, Elementary number theory, Divisibility, Consecutive Integers, Residue system, Constructive Proof.## 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.

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*Asian Journal of Fuzzy and Applied Mathematics*,

*2*(3). Retrieved from https://ajouronline.com/index.php/AJFAM/article/view/1331

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