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
4.3 Direct Proof and Counter Example III: Divisibility
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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