Dr. Cordero
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
- What does it mean for one integer to divide another?
- What is the definition of even number? Odd number?
- What can you say about the product of two odd numbers? Two even numbers? An odd and an even number?
- What can you say about the sum of two odd numbers? Two even numbers? An even and an odd number?
- 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?
- What is the converse of each statement in (e)? Is the converse of each statement true?
- 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?
- What is the greatest common divisor of two integers a and b?
- 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