4383_HW03_Sp12

4383_HW03_Sp12 - counterexample. a. If | p a and ( 29 2 2 |...

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Number Theory Homework #3 Sections 2.5, 3.1 and 3.2 1. Liz recently downloaded some songs and movies from her internet account. Songs cost $1.89 to download, movies cost $5.88 to download and Liz spent a total of $255.15. A. State all solutions to the linear Diophantine equation that models this problem. B. State the possible combinations of movies and songs Liz downloaded. 2. Section 2.5 Problem # 8(a). 3. Given two integers a and b with gcd(a, b) = p and p a prime, find all possible values of: a. ( 29 2 gcd , a b b. ( 29 2 2 gcd , a b c. ( 29 3 gcd , a b d. ( 29 3 2 gcd , a b 4. Let p be a prime number and a, b, c positive integers. For each of the following statements, state if it is true or false. If it is true, give a proof. If it is false, give at least one
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Unformatted text preview: counterexample. a. If | p a and ( 29 2 2 | + p a b , then | p b . b. If | , 1 n p a n , then | p a . c. If ( 29 2 2 | + p a b and ( 29 2 2 | + p b c , then ( 29 2 2 |-p a c . d. If ( 29 2 2 | + p a b and ( 29 2 2 | + p b c , then ( 29 2 2 | + p a c . 5. Section 3.1 Problem # 4 6. Section 3.1 Problem # 10 7. Section 3.1 Problem # 19 8. Does there exist a positive integer n such that n/2 is a perfect square, n/3 is a cube and n/5 is a fifth power? Justify your answer. 9. Section 3.2 Problem # 7 10. Section 3.2 Problem # 12...
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