MathSciDoc: An Archive for Mathematician ∫

This note presents two original and creative divisibility tests for the integers 2 and 5. Both rules use the factorial of the last digit of a number, combined with a small correction when the factorial value would disrupt the required modular congruence. The resulting tests are mathematically equivalent to the classical last-digit rules but reach the same conclusion through an unconventional factorial-based approach. Each theorem includes a short proof sketch, an explanation of the mechanism, and four worked examples. The methods are valid precisely because 2 and 5 divide the base 10, allowing the last digit to fully determine the remainder modulo of these divisors. The work is intended as an accessible piece of recreational number theory for educational purposes.

factorial divisibility test, last-digit congruence, recreational number theory