Math 5336 Number Theory                                                       Dr. Cordero

 

Class activities  for September 9th, 2002

Topics:

Mathematical Induction, II

Divisibility Theory in the Integers, I

 

I.Revisit work from last time:

 

1.      Hand-out #1   The Natural Number System

                          Problems 1-5.

 

2.      Hand-out #2  Induction and Recursion

·        Discuss the equivalence of the principle of Mathematical Induction and the Well-Ordering Principle.

·        Exercises # 3, 5, 6 a, 6c, 12 on p.159-160.

 

3.      Hand-out #3 Mathematical Induction

                                     Complete the proof on page 10.

 

4.      Additional Practice

       Textbook, Section 1.1, pp.6-7 # 1

 

 

 

II. Divisibility

 

1.      What does it mean for one integer to divide another?

2.      What is the definition of even number? Odd number?

3.      What can you say about the product of two odd numbers? Two even numbers? An odd and an even number?

4.      What can you say about the sum of two odd numbers? Two even numbers? An even and an odd number?

5.      Let a, b, d be integers. If d divides a and d divides b, is it necessarily true that d divides ab? What about a+b?

6.      What is the converse of each statement in (e)?  Is the converse of each statement true?

7.      Is it possible to find integers a, b and d such that d divides a+b and d divides a, but d doesn’t divide b?

8.      What is the greatest common divisor of two integers a and b?

9.      Theorem: 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.

 

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

10.  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.

 

          11.  Additional practice: §2.1 pp 19-20 # 1, 2, 3