Modular artithmetic
Modular arithmetic [ edit ] Main article: Modular arithmetic The hours on a clock form a group that uses addition modulo 12. Here, 9 + 4 ≡ 1 . Modular arithmetic for a modulus � defines any two elements � and � that differ by a multiple of � to be equivalent, denoted by � ≡ � ( mod � ) . Every integer is equivalent to one of the integers from 0 to � − 1 , and the operations of modular arithmetic modify normal arithmetic by replacing the result of any operation by its equivalent representative . Modular addition, defined in this way for the integers from 0 to � − 1 , forms a group, denoted as Z � or ( � / � � , + ) , with 0 as the identity element and � − � as the inverse element of � . A familiar example is addition of hours on the face of a clock , where 12 rather than 0 is chosen as the representativ...