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final-fall90

# final-fall90 - 165(b Use prime factorizations to compute...

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MAS 3300 - FINAL EXAM - FALL 1990 Wednesday December 19 Do ALL questions. All questions are of equal value. Write on ONE side of your paper. Calculators are allowed. Time allowed – TWO hours. 1. Show that if a R then a 0 = 0. 2. Use mathematical induction to prove that 1 + 2 + · · · + n = n ( n + 1) 2 , for all integers n 1. 3. Show that if a , b , c are integers and a | b and b | c then a | c . 4. Use unique factorization to prove that 2 is irrational. You may NOT assume Theorem (4.3). 5. (a) Use the Euclidean algorithm to find gcd(285
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Unformatted text preview: , 165). (b) Use prime factorizations to compute gcd(285 , 165). 6. Show that √ 2 6∈ Q [ √ 3]. 7. Show that if n is an integer, n ≥ 2, a ∈ Z n and a and n are relatively prime then a has a multiplicative inverse in Z n . 8. Find 13-1 in Z 23 . 9. For z 1 , z 2 ∈ C show that z 1 z 2 = z 1 z 2 . 10. Determine all solutions in C of z 8 = 1 . Write the solutions in rectangular form and simplify. 1...
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• Spring '08
• Staff
• Natural number, Greatest common divisor, Euclidean algorithm, Fundamental theorem of arithmetic, Euclidean domain

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