Discrete Structures

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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, 1601-1665. Definitions. If n and d are integers and d  0 then

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### 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, 1601-1665

4.3 Direct Proof and Counter Example III: Divisibility

Definitions

If n and d are integers and d 0 then

n is divisible bydiffn 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

NOTE
• Since the negation of an existential statement is universal, it follows that d does not divide niff, 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

Theorems
• Theorem 4.3.1 – 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 4.3.2 – Divisors of 1

The only divisors of 1 are a and -1.

• Theorem 4.3.3 – 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

Theorems
• Theorem 4.3.4 – Divisibility by a Prime

Any integer n > 1 is divisible by a prime number.

• Theorem 4.3.5 – Unique Factorization of Integers Theorem (Fundamental Theorem of Arithmetic)

4.3 Direct Proof and Counter Example III: Divisibility

Definition

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

Example – pg. 178 # 15
• Prove the statement directly from the definition of divisibility.

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

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

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

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