Math 5336 Number Theory Dr. Cordero

Class activities for September 23, 2002

Topic: Divisibility Theory in the Integers, III

- Discuss Test # 1
- 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?