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If we choose an arbitrary element of each residue class 3Z, 1 + nZ and 2 + nZ, we get a set of representatives for the residue classes Z/3Z (resp. e {0, 1, 2}, {0, 7, -4} . . There is an infinite number of possible complete sets of residues but a computer always uses {0, 1, 2, . . , n − 1} as a standard set. 3-5: The integers modulo n, denoted Zn , is the set of (equivalence classes of) integers {0, 1, 2, . . , n − 1}. Addition, substraction and multiplication in Zn are performed modulo n.

However, the integers 1 and 5 are relatively prime to 6. Thus, ϕ(6) = 2. For a formal definition of ϕ(n) with n ≥ 1, we write ϕ(n) = |{1 ≤ m ≤ n − 1 | gcd(n, m) = 1}|. The π(x)-function π(x) denotes the number of primes below x. For instance π(2) = 1, because we count 2 as the first prime, π(10) = 4, since there are four primes below 10, namely 2, 3, 5, and 7. 3 Number Theory 33 Property We now specify some properties of prime numbers without prooving them. 1. If n = k i=1 pei i and m = k i=1 pfi i , then gcd(n, m) = k i=1 min(ei ,fi ) pi .

The set M possesses the element 0 as identity element, and 3. every element a of M possesses an inverse (−a) so that a + (−a) = (−a) + a = 0. Subsequently M is a subgroup of the group (Z, +), because M ⊂ Z and (M, +) is a group. We now consider the order and some further properties of the group elements. 2-4: Let g be an element of the group G. The order of g is the least positive integer δ so that g δ = e, provided that such an integer exists. The order of g is defined to be infinite, if δ does not exist.