1. The essential quality of a proof is to compel belief.
Discrete Structures
Chapter 4: Elementary Number Theory and Methods of Proof
4.3 Direct Proof and Counter Example III: Divisibility
The essential quality of a proof is to compel belief.
– Pierre de Fermat,
4.3 Direct Proof and Counter Example III: Divisibility

2. 4.3 Direct Proof and Counter Example III: Divisibility
Definitions
If n and d are integers and d  0 then n is divisible by d iff n equals d times some integer. Instead of “n is divisible by d,” we can say that n is a multiple of d d is a factor of n d is a divisor of n d divides n The notation d | n is read “d divides n.” Symbolically, if n and d are integers and d  0. d | n   an integer k s.t. n = dk.
4.3 Direct Proof and Counter Example III: Divisibility

3. 4.3 Direct Proof and Counter Example III: Divisibility
NOTE
Since the negation of an existential statement is universal, it follows that d does not divide n iff, for all integers k, n  dk, or, in other words, n/d is not an integer.
4.3 Direct Proof and Counter Example III: Divisibility

4. 4.3 Direct Proof and Counter Example III: Divisibility
Theorems
Theorem – A Positive Divisor of a Positive Integer
For all integers a and b, if a and b are positive and a divides b, then a  b.
Theorem – Divisors of 1
The only divisors of 1 are a and -1.
Theorem – Transitivity of Divisibility
For all integers a ,b, and c, if a divides b and b divides c, then a divides c.
4.3 Direct Proof and Counter Example III: Divisibility

5. 4.3 Direct Proof and Counter Example III: Divisibility
Theorems
Theorem – Divisibility by a Prime
Any integer n > 1 is divisible by a prime number.
Theorem – Unique Factorization of Integers Theorem (Fundamental Theorem of Arithmetic)
4.3 Direct Proof and Counter Example III: Divisibility

6. 4.3 Direct Proof and Counter Example III: Divisibility
Definition
4.3 Direct Proof and Counter Example III: Divisibility

7. 4.3 Direct Proof and Counter Example III: Divisibility
Example – pg. 178 # 12
Give a reason for your answer. Assume that all variable represent integers.
4.3 Direct Proof and Counter Example III: Divisibility

8. 4.3 Direct Proof and Counter Example III: Divisibility
Example – pg. 178 # 16
Prove the statement directly from the definition of divisibility.
4.3 Direct Proof and Counter Example III: Divisibility

9. 4.3 Direct Proof and Counter Example III: Divisibility
Example – pg. 178 # 27
Determine whether the statement is true or false. Prove the statement directly from the definitions if it is true, and give a counterexample if it is false.
4.3 Direct Proof and Counter Example III: Divisibility

10. 4.3 Direct Proof and Counter Example III: Divisibility
Example – pg. 178 # 28
Determine whether the statement is true or false. Prove the statement directly from the definitions if it is true, and give a counterexample if it is false.
4.3 Direct Proof and Counter Example III: Divisibility

11. 4.3 Direct Proof and Counter Example III: Divisibility
Example – pg. 178 # 35
Two athletes run a circular track at a steady pace so that the first completes one round in 8 minutes and the second in 10 minutes. If they both start from the same spot at 4 pm, when will be the first they return to the start together.
4.3 Direct Proof and Counter Example III: Divisibility

12. 4.3 Direct Proof and Counter Example III: Divisibility
Example – pg. 178 # 37
Use the unique factorization theorem to write the following integers in standard factored form.
b. 5,733
c. 3,675
4.3 Direct Proof and Counter Example III: Divisibility