College Algebra Exam Review 32

College Algebra Exam Review 32 - 42 1 ALGEBRAIC THEMES...

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Unformatted text preview: 42 1. ALGEBRAIC THEMES Let’s look at a few examples of computations with congruences, or, equivalently, computations in Z n . The main principle to remember is that the congruence a b . mod n/ is equivalent to OEaŁ D OEbŁ in Z n . Example 1.7.8. (a) Compute the congruence class modulo 5 of 4 237 . This is easy be- cause 4 1 . mod 5/ , so 4 237 . 1/ 237 1 4 . mod 5/ . Thus in Z 5 , OE4 237 Ł D OE4Ł . (b) Compute the congruence class modulo 9 of 4 237 . As a strat- egy, let’s compute a few powers of 4 modulo 9. We have 4 2 7 . mod 9/ and 4 3 1 . mod 9/ . It follows that in Z 9 , OE4 3k Ł D OE4 3 Ł k D OE1Ł k D OE1Ł for all natural numbers k ; likewise, OE4 3k C 1 Ł D OE4Ł 3k OE4Ł D OE4Ł , and OE4 3k C 2 Ł D OE4Ł 3k OE4Ł 2 D OE7Ł . So to compute 4 237 modulo 9, we only have to find the conjugacy class of 237 modulo 3; since 237 is divisible by 3, we have OE4 237 Ł D OE1Ł in Z 9 ....
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