Math 5336 Number Theory                                                       Dr. Cordero

 

Class activities for September 23, 2002

Topic: Divisibility Theory in the Integers, III

 

  1. Discuss Test # 1
  2. Discuss the homework questions:

        Practice: 2.1 pp 19-20 # 1, 2, 3a, 3b.

        (Part 5) What is the statement for integers that are relatively prime?

       3.  Continue with handout from last time:

                       6. a. Question: If a|b and b|c, must ab|c? Study some cases.

   b. Prove the following theorem:

        Theorem If a|c and b|c with gcd(a,b)=1, then ab|c.

 

7. a. Question: If a|bc, must a|b and a|b?

     b. If a|bc with gcd(a,b)=1, must a|c?

 

8. Practice: 2.2 p.25 # 7, 14

 

9. Euclidean Algorithm: Read page 27 (up to Lemma ).

        What is the Euclidean Algorithm?

        What is the Euclidean Algorithm used for?

        Write an example where you use the Euclidean Algorithm.

 

10. a. Find gcd(364, 140).

b. How can we write gcd(364, 140) as a linear combination of           364 and 140?

 

11. Use the Extended Euclidean Algorithm to find integers x and y such that:

        141x+120y=3

        243x+41y=1

        243x+41y=6

        1721x+378y=7

 

12. The Stamps Problem

  Suppose you have an unlimited supply of 6 -cents stamps and 11- cents stamps.

What amounts of postage can you make with these stamps?

        Solve the Stamps problem.

        What if the stamps were of 6-cents and 9-cents?

        What if the stamps were of 6-cents and 10-cents?