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test2-spring97

# test2-spring97 - A 5[6 pts Prove that Z does not have an...

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TEST 2 - MAS 3300 - Spring 1997 March 28, 1997 There are 9 questions giving a total of 50 points. 100% = 45 points Work submitted must be written in a proper and coherent fashion. Write on ONE side of your paper. 1 . [4 + 4 = 8 pts] (a) Use the Euclidean algorithm to find gcd(285 , 165). (b) Use the prime factorizations to compute gcd(285 , 165). 2 . [6 pts] Prove that there are infinitely many primes. HINT : Suppose there are only finitely many primes say p 1 , p 2 , . . . p m . 3 . [5 pts] Use unique factorization to prove that 2 is irrational. 4 . [3 pts] Let b R and suppose A is a non-empty set of real numbers. Define what it means for b to be an upper bound for the set
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Unformatted text preview: A . 5 . [6 pts] Prove that Z does not have an upper bound. HINT : Use LUB. 6 . [5 pts] Let α = 2 1 / 3 . Deﬁne Q [ α ] := { a + bα + cα 2 : a,b,c ∈ Q } . Prove that Q [ α ] is closed under multiplication. 7 . [6 pts] Prove that if a ∈ Z n and a and n are relatively prime then a has a multiplicative inverse in Z n . 8 . [5 pts] Find the multiplicative inverse of 19 in Z 81 . 9 . [2 + 2 + 2 = 6 pts] (a) Deﬁne the term ordered ﬁeld . (b) Give an example of an ordered ﬁeld. (c) Give an example of a ﬁeld which is not an ordered ﬁeld. 1...
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