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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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