College Algebra Exam Review 24

College Algebra Exam Review 24 - by Proposition 1.6.2 We...

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34 1. ALGEBRAIC THEMES (Note that n=q 1 could be 1 and one or both of r ± 1 and s ± 1 could be 0.) Since n=q 1 < n 1 , it follows from the induction hypothesis that r D s and q i D p i for all i ² 2 . n How do we actually compute the prime factorization of a natural num- ber? The conceptually clear but computationally difficult method that you learned in school for factoring a natural number n is to test all natural num- bers no larger than p n to see if any divides n . If no factor is found, then n must be prime. If a factor a is found, then we can write n D a ³ .n=a/ and proceed to search for factors of a and n=a . We continue this proce- dure until only prime factors appear. Unfortunately, this procedure is very inefficient. Better methods are known, but no truly efficient methods are available for factoring very large natural numbers. The greatest common divisor of several integers Definition 1.6.19. A natural number ˛ is the greatest common divisor of nonzero integers a 1 ;a 2 ;:::;a n , if (a) ˛ divides each a i and (b) whenever ˇ 2 N divides each a i , then ˇ also divides ˛ . Notice that if the greatest common divisor is unique, if it exists at all,
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Unformatted text preview: by Proposition 1.6.2 . We prove existence by means of a generalization of the matrix formulation of the Euclidean algorithm. Lemma 1.6.20. Given nonzero integers a 1 ;a 2 ;:::;a n ( n ² 2 ), there is a natural number d and an n –by– n integer matrix Q such that Q is invert-ible, Q ± 1 also has integer entries, and .d;0;:::;0/ D .a 1 ;a 2 ;:::;a n /Q: Proof. We proceed by induction on n . The base case n D 2 is established in the proof of Proposition 1.6.10 . Fix n > 2 and suppose the assertion holds for lists of fewer than n nonzero integers. Then there exists a natural number d 1 and a n ± 1 –by– n ± 1 integer matrix Q 1 with integer inverse such that .d 1 ;0;:::;0/ D .a 2 ;:::;a n /Q 1 : By the base case n D 2 , there is a natural number d and a 2 –by– 2 integer matrix Q 2 with integer inverse such that .d;0/ D .a 1 ;d 1 /Q 2 :...
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