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Past Exam for MATH 453 - Elementary Theory of Numbers at Illinois (UIUC)

Exam Information

Material Type:Final
Professor:Staff
Class:MATH 453 - Elementary Theory of Numbers
Subject:Mathematics
University:University of Illinois - Urbana-Champaign
Term:Spring 2008
Keywords:
  • Incongruent Quadratic Nonresidues
  • Prime Factor Congruent
  • Moebius Inversion Formula
  • Quadratic Nonresidue
  • Exam Solutions Spring
  • Incongruent Integers
  • Combination
  • Explanation
  • Justification
  • Definitions
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Sample Document Text

Math 453, Section X13 Final Exam Solutions Spring 2008 Problem 1 (15 points) (True/false questions) For each of the following statements, say if it is true or false, and provide a brief justi- fication for your claim. Credit on these questions is based on your justification. A simple true/false answer, without justification, or with an incorrect justification, won't earn credit. For true statements, a justification typically consists of citing and applying an appropriate theorem, if nec- essary stating why the cited theorem can be applied. Be specific; e.g., say "Since (453,347) = 1, Euler's Theorem with a = 453 and b = 347 applies and guarantees the existence of a solution ..." rather than some- thing like "true by Euler's Theorem". For false statements, usually a specific counterexample may be enough. Note, however, that a different strategy is required to disprove statements asserting something for infinitely many (rather than all) integers. (i) There exist infinitely many solutions x,y ? Z to t...

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