40 1. ALGEBRAIC THEMES We denote by Z n the set of residue classes modulo n . The set Z n has a natural algebraic structure which we now describe. Let A and B be elements of Z n , and let a 2 A and b 2 B ; we say that a is a representative of the residue class A , and b a representative of the residue class B . The class Œa C bŁ and the class ŒabŁ are independent of the choice of representatives. For if a0 is another representative of A and b0 another representative of B , then a ± a0 . mod n/ and b ± b0 . mod n/ ; therefore a C b ± a0 C b0 . mod n/ and ab ± a0 b0 . mod n/ according to Lemma 1.7.5 . Thus Œa C bŁ D Œa0 C b0 Ł and ŒabŁ D Œa0 b0 Ł . This means that it makes sense to deﬁne A C B D Œa C bŁ and AB D ŒabŁ . Another way to write these deﬁnitions is ŒaŁ C ŒbŁ D Œa C bŁ; ŒaŁŒbŁ D ŒabŁ: (1.7.1) Example 1.7.6. Let us look at another example in which we cannot deﬁne operations on classes of numbers in the same way (in order to see what the issue is in the preceding discussion). Let
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