Math 5336 Number Theory Dr. Cordero
Class activities for September 16, 2002
Topic: Divisibility Theory in the Integers, II
1. Question: Let a, b, c be integers. If a|b and a|c, must a|bx+cy for any integers x and y?
2. Greatest common divisor
What is the greatest common divisor of two integers a and b?
· gcd(14, 21)
· gcd(15, 35)
· gcd(13, 43)
· gcd(7696, 4144)
3. The division algorithm
Let a and b be integers with b = 0. Then there are unique integers q and r such that 0<r<|b| and a=bq+r.
a. Practice: Find the unique q and r guaranteed by the Division Algorithm for each pair a and b:
· a=96, b=48
· a=96, b=36
· a=87, b=15
· a=7696, b=4144
· , b=47
b. Applications of the Division Algorithm
· Use the division algorithm to show that the square of any odd integer n is of the form 8k+1.
· Use the division algorithm to show that for any , the expression is an integer.
c. Practice: §2.1 pp 19-20 # 1, 2, 3a, 3b.
4. Question: What does it mean for two integers a and b to be “relatively prime”?
· Give an example of a pair of integers a and b which are relatively prime but none of which is a prime number.
5. Study the following theorem:
Theorem Given integers a and b not both 0, there exists integers x and y such that gcd(a,b)=ax+by.
· What is the statement for integers that are relatively prime?
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:
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?