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.

n and d , d | n 
n
is not an integer.
d
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 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
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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)

G iven any integer n  1,  k 
, distinct prim e n um bers
p1 , p 2 , p 3 , ..., p n , and positive integers e1 , e 2 , e 3 , ..., e n s .t.
n  p1 1 p 2 2 p 3 3
e
e
e
e
p nn
and any other expression for n as a pr oduct of prim e num bers
is identical to this except, perhaps, fo r the order in w hich the
factors are w ritten.
4.3 Direct Proof and Counter Example III:
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Definition
G iven any integer n  1, the standar d factored form of n is an
expression of the form
n  p1 1 p 2 2 p 3 3
e
e
e
e
pkk
w here k is a positive integer; p1 , p 2 , p 3 , ..., p k are prim e num bers;
e1 , e 2 , e 3 , ..., e k are positive integers; and p1  p 2  p 3  ...  p k .
4.3 Direct Proof and Counter Example III:
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Example – pg. 178 # 12
• Give a reason for your answer. Assume that all
variable represent integers.
If n  4 k  1, does 8 divide n  1?
2
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 a , b , and c , if a | b and a | c
then a |  b  c  .
4.3 Direct Proof and Counter Example III:
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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 a , b , and c , if a |  b  c  then a | b o r 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.
F o r all in teg ers a , b , an d c , if a |  b c  th en a | b o r a | c .
4.3 Direct Proof and Counter Example III:
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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
4.3 Direct Proof and Counter Example III:
Divisibility
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