How does a 24-hour clock demonstrate arithmetic modulo 24?

A 24-hour clock serves as a practical example of modular arithmetic because it operates on a count that resets itself to zero once the modulus, $N=24$, is reached.1

In this system, the clock does not continue counting indefinitely; instead, it treats time as a cycle. When the time reaches 24 hours, it resets to 0 at midnight, effectively mapping all possible time values to the set of residues $\{0,1,2,\dots ,23\}$.1

This demonstrates the core principle of modular arithmetic, where numbers are replaced by their remainders after division by a fixed modulus. Just as the clock wraps around after 24 hours, modular arithmetic performs calculations by taking the sum of numbers and reducing them to a value between 0 and $N-1$ by subtracting the modulus as many times as necessary. This cyclical nature is a fundamental concept in number theory, providing a framework for solving problems involving repeating patterns and cycles.213