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?