College Algebra Exam Review 22

College Algebra Exam Review 22 - 1 Proposition 1.6.13 Two...

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32 1. ALGEBRAIC THEMES Example 1.6.11. Find the greatest common divisor of 1734282 and 452376. Successive divisions with remainder give 1734282 D 3 ± 452376 C 377154 452376 D 377154 C 75222 377154 D 5 ± 75222 C 1044 75222 D 72 ± 1044 C 54 1044 D 19 ± 54 C 18 54 D 3 ± 18: Thus 18 D g : c : d :.1734282;452376/ . We can find the coefficients s;t such that 18 D s 1734282 C t 452376: The sequence of quotients q 1 ;q 2 ;:::;q 6 in the algorithm is 3;1;5;72;19;3 . The q k determine matrices Q k D ± 0 1 1 ² q k ² . The coefficients s;t com- prise the first column of Q D Q 1 Q 2 ³³³ Q 6 . The result is 18 D 8233 ± 1734282 ² 31563 ± 452376: Definition 1.6.12. Nonzero integers m and n are relatively prime if g : c : d :.m;n/ D
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Unformatted text preview: 1 . Proposition 1.6.13. Two integers m and n are relatively prime if, and only if, 1 2 I.m;n/ . Proof. Exercise 1.6.8 . n Example 1.6.14. The integers 21 and 16 are relatively prime and 1 D ² 3 ± 21 C 4 ± 16 . Proposition 1.6.15. If p is a prime number and a is any nonzero integer, then either p divides a or p and a are relatively prime. Proof. Exercise 1.6.9 . n From here, it is a only a short way to the proof of uniqueness of prime factorizations. The key observation is the following:...
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.

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