MC36 -- Relation-Based Structures -- Fall 1998

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Assignments for Week 7

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Problems Set #6  - html   postscript

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Tuesday, October 27:

•  We will discuss portions of Grimaldi:  4.3
•  Try proving the parts of Theorem 4.3 that are left to the reader.  We will use (e) and (f) often.
• Note that  Theorem 4.5, The Division Algorithm, is proved for b>0.  Problem #25 gives the general case.
• Look at Example 4.22 carefully.   How does the division algorithm work when "a" is negative?
• Representing integers in bases other than 10.

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Thursday, October 29:

• Read Grimaldi 4.4
• Notice that implicit in the proof of Theorem 4.6 is the fact that the greatest common divisor of two positive integers can be written as a linear combination of the integers.  (gcd(a,b)=as+tb, where s, t are integers)  Note that s and t need not be positive nor are they unique.
• How does the Euclidean algorithm work?  Why does it work?
• How can we use the Euclidean algorithm to express gcd(a,b) as a linear combination of a and b?  Pay close attention to Example 4.31, especially the paragraph on the top of p228.  What is he doing there?

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Friday, October 30:  Modular arithmetic and a little cryptography

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Saturday, October 31:  Halloween.