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final-spring97 - Name MAS 3300 Numbers and Polynomials...

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Name: MAS 3300 Numbers and Polynomials Spring 1997 FINAL EXAM Instructions: There are 12 questions. Do only TEN questions. With TEN complete problems there are 100 total points. Write on ONE side of the paper provided. Show all necessary working and reasoning to receive full credit. Your work needs to be written in a proper and coherent fashion. When giving proofs your reasoning should be clear. Calculators are allowed. You may answer questions 4(ii), and 12(i) on the Test Paper. A table of primes is supplied. PLEASE GRADE THE FOLLOWING TEN QUESTIONS: 1
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1. [10 points] Let a R . Prove a 0 = 0 . 2. [10 points] Let a , b R and suppose a < 0 and b < 0. Prove ab > 0 . 3. [10 points] Use mathematical induction to prove 1 2 + 2 2 + 3 2 + · · · + n 2 = 1 6 n ( n + 1)(2 n + 1) , for all integers n 1. 4. [1 + 1 + 8 = 10 points] (i) Define what it means for two integers to be relatively prime . (ii) Complete the following Theorem . Suppose a , b Z . Then a and b are relatively prime iff there exist integers x and y such that .............................................................................. (iii) Let a , b , c Z . Using (ii), prove the following Theorem . Let a , b , c Z . If a | bc and a and b are relatively prime then a | c . 5. [10 points] Compute the gcd of 525 and 595 (i) using the Euclidean algorithm;
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