4.3 Direct Proof and Counter Example III: Divisibility

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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

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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

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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.

and , |n d d is not an integer.n

nd

4.3 Direct Proof and Counter Example III: Divisibility

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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 1The 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

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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)

31 2

1 2 3 1 2 3

1 2 3

Given any integer 1, , distinct prime numbers

, , ,..., , and positive integers , , ,..., s.t.

and any other expression for as a pr

n

n n

e ee en

n k

p p p p e e e e

n p p p p

n

oduct of prime numbers

is identical to this except, perhaps, for the order in which the

factors are written.

4.3 Direct Proof and Counter Example III: Divisibility

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Definition

31 21 2 3

1 2 3

1

Given any integer 1, the of is an

expression of the form

where is a positive integer;

standar

, , ,..., are prime numbers;

d factored form

ke ee ek

k

n n

n p p p p

k p p p p

e

2 3 1 2 3, , ,..., are positive integers; and ... .k ke e e p p p p

4.3 Direct Proof and Counter Example III: Divisibility

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Example – pg. 178 # 12

• Give a reason for your answer. Assume that all variable represent integers.

2If 4 1, does 8 divide 1?n k n

4.3 Direct Proof and Counter Example III: Divisibility

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Example – pg. 178 # 16

• Prove the statement directly from the definition of divisibility.

For all integers , , and , if | and |

then | .

a b c a b a c

a b c

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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.

For all integers , , and , if | then | or | .a b c a b c a b a c

4.3 Direct Proof and Counter Example III: Divisibility

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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.

For all integers , , and , if | then | or | .a b c a bc a b a c

4.3 Direct Proof and Counter Example III: Divisibility

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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

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Example – pg. 178 # 37

• Use the unique factorization theorem to write the following integers in standard factored form.– b. 5,733– c. 3,675