Section 2.1
Arithmetic modulo n

Clock arithmetic is an example of arithmetic modulo an integer, in this case 24. If the time is 15.00 hours and 20 hours pass by, then it is 11.00 hours (15 + 20 equals 11 modulo 24). If 83 hours elapse, it is 2 o'clock in the morning (15 + 83 equals 2 modulo 24). We look at time within multiples of 24.

In this section we will formalize arithmetic modulo an integer.

Definition

Let n be an integer. On the set Z we define the relation `congruent modulo n' as follows:

a and b are congruent modulo n if and only if

n | a - b.

If a and b are congruent modulo n, we often denote this by

a = b mod n.

The relation `congruent modulo n' is an equivalence relation. Using the notation

k + nZ = {k + nm | m in Z},

we find (for positive n) exactly n distinct equivalence classes:

nZ , 1 + nZ , ..., n - 1 + nZ.

In words: the multiples of n, the multiples of n plus 1, ..., the multiples of n plus n - 1.

The class containing the integer m (i.e., m + nZ) will often be denoted by m (mod n). The number m is a representative of this equivalence class. If no confusion arises, we will also denote the class of m modulo n by m itself.

The set of equivalence classes of Z modulo n is denoted by Z/nZ.