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Exam1Review - (b Use your work from part(a to write the...

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Math 302 Modern Algebra Spring 2009 Exam 1 Information Date: Friday, February 27 Covering: Chapter 1, Sections 1, 2, 3, 4 Chapter 2, Sections 1, 2, 3 General Comments You will need to bring a large (8 1 2 × 11) blue book to the exam. You should also bring the Theorem Sheet to the exam. The exam problems for Exam 1 (as well as all the exams) will be split approximately 50–50 between general proofs and problems that are computationally oriented. Exam 1 Review Problems 1. Prove Theorem 1.10 for positive integers: Every positive integer greater than 1 is the product of primes. [ Hint: Apply the Well-Ordering Axiom with S the set of all positive integers greater than 1 that are not the product of primes.] 2. (a) Find the greatest common divisor of 22 and 291 using the Euclidean Algorithm.
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Unformatted text preview: (b) Use your work from part (a) to write the greatest common divisor as a linear combination of 22 and 291. (c) Solve the equation 291 x = 5 in Z 22 . 3. Let p be an integer other than 0, ± 1 with this property: For all integers b and c , if p | bc , then p | b or p | c . Prove that p is prime. 4. Prove: For all integers a and b and all positive integers n , a ≡ b (mod n ) if and only if a and b have the same remainder when divided by n . 5. Let a and n be integers with n > 1. Prove: If ( a, n ) = 1 in Z , then the equation [ a ] ⊙ x = [1] has a solution in Z n . (This is the result of Corollary 2.10. You should not refer to that corollary in your proof.)...
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