It is of common experience that a railway time table is fixed with the prevision of 24 hours in a day and night. When we say that a particular train is arriving at 15 hours, it implies that the train will arrive at 3 p.m. according to our watch. Thus all the timing starting from 12 to 23 hours correspond to one of 0, 1, 3,…, 11 O’clock, as indicated on watches. In other words, all integers from 12 to 23 are equivalent to one or the other of integers 0, 1, 2, 3, …, 11 with modulo 12. Thus, the integers in question are divided into 12 classes.
In the manner described above the integer could be divided into 2 classes, or 5 classes or m (m being a positive integer) classes, and then we would have written mod 2 or mod 5 or mod m. This system of representing integers is called a modulo system.