College Algebra Exam Review 27

College Algebra Exam Review 27 - 1.7 MODULAR ARITHMETIC...

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1.7. MODULAR ARITHMETIC 37 1.6.13. (a) Write a program in your favorite programming language to com- pute the greatest common divisor of two nonzero integers, using the approach of repeated division with remainders. Get your pro- gram to explicitly give the greatest common divisor as an integer linear combination of the given nonzero integers. (b) Another method of finding the greatest common divisor would be to compute the prime factorizations of the two integers and then to take the largest collection of prime factors common to the two factorizations. This method is often taught in school math- ematics. How do the two methods compare in computational efficiency? 1.6.14. (a) Let I D I.n 1 ; n 2 ; : : : ; n k / D f m 1 n 1 C m 2 n 2 C : : : m k n k W m 1 ; : : : ; m k 2 Z g : Show that if x; y 2 I , then x C y 2 I and x 2 I . Show that if x 2 Z and a 2 I , then xa 2 I . (b) Show that g : c : d .n 1 ; n 2 ; : : : ; n k / is the smallest element of I \ N . 1.6.15. (a) Develop an algorithm to compute g : c : d .n 1 ; n 2 ; : : : ; n k / .
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