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Section 5.1
Number
Theory
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
What You Will Learn
Introduction to Number Theory
Prime Numbers
Composite Numbers
Prime Factorization
5.1-2
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Number Theory
The study of numbers and their
properties.
The numbers we use to count are
called counting numbers, or natural
numbers, denoted by N.
N = {1, 2, 3, 4, 5, …}
5.1-3
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Factors
The natural numbers that are
multiplied together are called factors
of the product.
A natural number may have many
factors.
The factors of 18 are 1, 2, 3, 6, 9 and
18.
5.1-4
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Divisors
If a and b are natural numbers, we say
that a is a divisor of b or a divides b,
symbolized a|b if the quotient of b
divided by a has a remainder of 0.
If a divides b, then b is divisible by a.
5.1-5
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Prime and Composite Numbers
A prime number is a natural number
greater than 1 that has exactly two
factors (or divisors), itself and 1.
A composite number is a natural
number that is divisible by a number
other than itself and 1.
The number 1 is neither prime nor
composite; it is called a unit.
5.1-6
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Divisibility Rules
Divisible
by
5.1-7
Test
Example
2
The number is even.
924; even
3
The sum of the digits of
924; 8 + 4 + 6 =
the number is divisible by 18 and 18 is
3.
divisible by 3
4
The number formed by
the last two digits of the
number is divisible by 4.
924; 24
is divisible by 4
5
The number ends in 0 or
5.
265; ends in 5
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Divisibility Rules
Divisible
by
Example
6
The number is divisible
by both 2 and 3.
8
The number formed by
5824;
the last three digits of
824 is divisible by 8
the number is divisible by
8.
The sum of the digits of
837; 8 + 3 + 7 =
the number is divisible by 18 and 18 is
9.
divisible by 9
9
10
5.1-8
Test
The number ends in 0.
924; divisible by
both 2 and 3
730; ends in 0
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
The Fundamental Theorem of
Arithmetic
Every composite number can be
expressed as a unique product of
prime numbers.
This unique product is referred to as
the prime factorization of the number.
5.1-9
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Finding Prime Factorizations
Method 1: Branching
Select any two numbers whose product
is the number to be factored.
If the factors are not prime numbers,
continue factoring each number until
all numbers are prime.
5.1-10
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 2: Prime Factorization
by Branching
Write 1500 as a product of primes.
Solution
or
Thus, the prime factorization of
1500 = 2 • 2 • 3 • 5 • 5 • 5 = 22 • 3 • 53
5.1-11
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Method 2: Division
1. Divide the given number by the
smallest prime number by which it is
divisible.
2. Place the quotient under the given
number.
3. Divide the quotient by the smallest
prime number by which it is divisible
and again record the quotient.
4. Repeat this process until the
quotient is a prime number.
5.1-12
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 2: Prime Factorization
by Division
Write 1500 as a product of prime
numbers.
2 1500
Solution
2 750
3 375
5 125
5 25
5
1500 = 2 • 2 • 3 • 5 • 5 • 5 = 22 • 3 • 53
5.1-13
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Greatest Common Divisor
The greatest common divisor (GCD)
of a set of natural numbers is the
largest natural number that divides
(without remainder) every number in
that set.
5.1-14
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To Determine the GCD of Two or
More Numbers
1. Determine the prime factorization of
each number.
2. List each prime factor with smallest
exponent that appears in each of
the prime factorizations.
3. Determine the product of the
factors found in step 2.
5.1-15
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Example 4: Using Prime
Factorization to Determine the GCD
Determine the GCD of 54 and 90.
Solution
54 = 2 • 33
90 = 2 • 32 • 5
Prime factors with smallest exponents
that appear in each factorization
2 and 32
The GCD is 2 • 32 = 18.
5.1-16
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Least Common Multiple
The least common multiple (LCM) of
a set of natural numbers is the
smallest natural number that is
divisible (without remainder) by each
element of the set.
5.1-17
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To Determine the LCM of Two or
More Numbers
1. Determine the prime factorization of
each number.
2. List each prime factor with the
greatest exponent that appears in
any of the prime factorizations.
3. Determine the product of the factors
found in step 2.
5.1-18
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Example 6: Using Prime
Factorization to Determine the LCM
Determine the LCM of 54 and 90.
Solution
54 = 2 • 33
90 = 2 • 32 • 5
Prime factors with greatest exponents
that appear in either factorization
2, 33 and 5
The LCM is 2 • 33 • 5 = 90.
5.1-19
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Search for Larger Prime Numbers
More than 2000 years ago, the Greek
mathematician Euclid proved that
there is no largest prime number.
Mathematicians, however, continue to
strive to find larger and larger prime
numbers.
5.1-20
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Mersenne Primes
Marin Mersenne (1588–1648), a
seventeenth-century monk, found that
numbers of the form 2n – 1 are often
prime numbers when n is a prime
number
Numbers of the form 2n – 1 are
referred to as Mersenne primes
5.1-21
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Mersenne Primes
The first 10 Mersenne primes occur
when
n = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89
The first time the expression does not
generate a prime number, for prime
number n, is when n is 11
5.1-22
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Mersenne Primes
Largest prime discovered August 23,
2008 by Edson Smith, UCLA
243,112,609 – 1
It is 12,978,189 digits long
Written in 12-point font it is 17 miles
long
5.1-23
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Fermat Numbers
Pierre de Fermat (1601 – 1665)
conjectured that each number of the
2n
form 2  1, now referred to as a
Fermat number, was prime for each
natural number n.
A conjecture is a supposition that has
not been proved or disproved.
5.1-24
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Fermat Numbers
In 1732, Leonhard Euler proved that
for n = 5, 232 + 1 was a composite
number, thus disproving Fermat’s
conjecture.
Since Euler’s time, mathematicians
have been able to evaluate only the
sixth, seventh, eighth, ninth, tenth,
and eleventh Fermat numbers and
each of these numbers has been shown
to be composite.
5.1-25
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Goldbach’s Conjecture
In 1742, Christian Goldbach
conjectured that every even number
greater than or equal to 4 can be
represented as the sum of two (not
necessarily distinct) prime numbers.
For example, 4 = 2 + 2, 6 = 3 + 3, 8
= 3 + 5, 10 = 5 + 5, 12 = 5 + 7.
It remains unproven to this day.
5.1-26
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Twin Prime Conjecture
Twin primes are primes of the form p
and p + 2 (3 and 5, 11 and 13).
The conjecture states that there are an
infinite number of pairs of twin primes.
The largest known twin primes are of
the form 65,516,468,355 2333,333 ± 1.
The were found by the efforts of two
research groups: Twin Prime Search
and PrimeGrid on August 6, 2009.
5.1-27
Copyright 2013, 2010, 2007, Pearson, Education, Inc.