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College Algebra Exam Review 25

# College Algebra Exam Review 25 - c d a 1 a n as well as...

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1.6. DIVISIBILITY IN THE INTEGERS 35 Then .d; 0; : : : ; 0/ D .a 1 ; a 2 ; : : : ; a n / 2 6 6 6 4 1 0 0 0 : : : 0 Q 1 3 7 7 7 5 2 6 6 6 6 6 4 Q 2 0 0 0 0 0 0 : : : : : : 0 0 E 3 7 7 7 7 7 5 where E denote the n 2 –by– n 2 identity matrix. n Proposition 1.6.21. The greatest common divisor of nonzero inte- gers a 1 ; a 2 ; : : : ; a n exists, and is an integer linear combination of a 1 ; a 2 ; : : : ; a n . Proof. The natural number d in the lemma is an integer linear combination of a 1 ; a 2 ; : : : ; a n , since .d; 0; : : : ; 0/ D .a 1 ; a 2 ; : : : ; a n /Q; and is a common divisor of a 1 ; a 2 ; : : : ; a n since .a 1 ; a 2 ; : : : ; a n / D .d; 0; : : : ; 0/Q 1 : It follows that d is the greatest common divisor of a 1 ; : : : ; a n . n The greatest common divisor of a 1 ; : : : ; a n is denoted g : c : d . a 1 ; : : : ; a n / . The proof of the proposition can be converted into an algorithm for com-
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Unformatted text preview: c : d . a 1 ;:::; a n / as well as integers s 1 ;s 2 ;:::;s n such g : c : d .a 1 ;:::;a n / D s 1 a 1 C s 2 a 2 C ±±± C s n a n : Exercises 1.6 1.6.1. Complete the following sketch of a proof of Proposition 1.6.7 using the well–ordering principle. (a) If a ¤ , consider the set S of nonnegative integers that can be written in the form a ² sd , where s is an integer. Show that S is nonempty. (b) By the well–ordering principle, S has a least element, which we write as r D a ² qd . Then we have a D qd C r . Show that that r < d ....
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