How does a circular protractor demonstrate arithmetic modulo 360?

A circular protractor demonstrates arithmetic modulo 360 because it functions as a system where the count resets itself to zero once the modulus, $N=360$, is reached.1

In modular arithmetic, values are restricted to a set of residues ranging from $0$ to $N-1$. On a circular protractor, moving past the 359-degree mark returns the measurement to 0 degrees, effectively completing a cycle. This cyclic nature mirrors how modular arithmetic operates in number theory, where calculations are performed within a finite set of congruence classes. Just as a 24-hour clock resets at midnight, the protractor’s scale ensures that any angle measurement is equivalent to its remainder when divided by 360.1