What is special about the divisors of 24?

TL;DR

This paper explores a unique number theoretic characterization of the divisors of 24 using modular multiplication tables, supported by multiple diverse proofs from different mathematical techniques.

Contribution

It introduces a novel characterization of the divisors of 24 and demonstrates it through five distinct proofs employing various advanced number theory methods.

Findings

Characterization of divisors of 24 via modular multiplication tables

Five different proofs using diverse number theory techniques

Insight into the structure of divisors of 24 through multiple perspectives

Abstract

What is an interesting number theoretic or a combinatorial characterization of the divisors of 24 amongst all positive integers? In this paper I will provide one characterization in terms of modular multiplication tables. This idea evolved interestingly from a question raised by a student in my elementary number theory class. I will give the characterization and then provide 5 proofs using various techniques: Chinese remainder theorem, structure theory of units, Dirichlet's theorem on primes in an arithmetic progression, Bertrand-Chebyshev theorem, and results of Erdos and Ramanujan on the pi(x) function.