What is special about the divisors of 24?

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.