Divisibility rules for integers presented as permutations

TL;DR

This paper introduces a novel approach to divisibility rules for integers by representing them as permutations using factoradic notation, providing explicit formulas for modular arithmetic.

Contribution

It presents a new permutation-based representation of integers and derives divisibility rules directly from factoradic digits, offering a fresh perspective beyond traditional methods.

Findings

Derived formulas for n mod k using factoradic digits

Established new divisibility rules based on permutation representations

Enhanced understanding of integer divisibility through permutation structures

Abstract

In this note, we represent integers in a type of factoradic notation. Rather than use the corresponding Lehmer code, we will view integers as permutations. Given a pair of integers n and k, we give a formula for n mod k in terms of the factoradic digits, and use this to deduce various divisibility rules.