Z/nZ is the integers modulo n.
so, for example, Z/2Z, = {[0], [1]}
where [0] = {....,-6,-4,-2,0,2,4,6,...} = 2Z, the set of even integers.
[1] = {....,-5,-3,-1,1,3,5,7,....} = 1 + 2Z, the set of odd integers.
in general [k] is the set of all integers congruent to k modulo n.
[k] is an equivalence class, as congruence modulo n is an equivalence relationship.
these equivalence classes can be given an arithmetic structure, because:
[k] + [m] = [k+m], and [k][m] = [km].
the notation Z/nZ is group-theoretic, for any integer n, (nZ,+) is a subgroup of the additive group of the integers, (Z,+). since (Z,+) is an abelian or commutative group, nZ is normal, which is to say:
k+nZ = {k + nr, k,r in Z} = Zn+k = {nr + k,k,r in Z}
note that k+nZ is just another way of writing [k].
this fact allows us to define [k] + [m], because
[k] + [m] = (k + nZ) + (m + nZ) = k + (nZ + m) + nZ = k + (m + nZ) + nZ
= (k + m) + nZ + nZ = (k + m) + nZ
since nZ is a subgroup of Z, any sum of multiples of n is again a multiple of n, so
nZ + nZ = nZ, and (k + m) + nZ + nZ = (k+m) + nZ = [k+m].
it is important to realize that k is a number, and [k] is a set.
the elements of Z/nZ are the sets [0], [1],...,[n-1]. sometimes these are called the co-sets of nZ in Z.
when no confusion can arise, sometimes these are just written:
0,1,....,n-1. be careful about this.
it is important to realize the congruence class of an integer k modulo n isn't quite the same thing as the integer k.
Z/pZ is the same thing as Z/nZ, except p is taken to be a prime number. these are somewhat special because they can be given the structure of a field (that is, you can add, subtract, multiply and divide in them), whereas in Z/nZ, you can only have the structure of a ring.
here is Z/4Z: {[0],[1],[2],[3]}
in Z/4Z, [2][2] = [4] = [0].
that means [2] is a zero-divisor, it doesn't have a multiplicative inverse.in Z/4Z, dividing by [2] is just as bad as dividing by 0 is in Z.
note that [3][3] = [9] = [4*2 + 1] = [4][2] + [1] = [0][2] + 1 = [0] + [1] = [1], so you CAN divide by [3] in Z/4Z: 1/[3] = [3].
Z/nZ acts a little stranger than the integers we are used to, but it is useful for proving things about divisibility in the normal integers. for example:
Z/mZ ∩ Z/nZ = Z/rZ, where r is the least common multiple of m and n.