Download § 4-2 Prime and Composite Numbers

§ 4-2 Prime and Composite Numbers
Definitions
What makes a whole number prime?
Definitions
What makes a whole number prime?
Definition
A prime number is a whole number that has exactly two distinct
factors, one and itself.
Definitions
What makes a whole number prime?
Definition
A prime number is a whole number that has exactly two distinct
factors, one and itself.
If a number is not prime, then it is ...
Definition
A composite number is a whole number that has more than two
distinct factors.
Definitions
What makes a whole number prime?
Definition
A prime number is a whole number that has exactly two distinct
factors, one and itself.
If a number is not prime, then it is ...
Definition
A composite number is a whole number that has more than two
distinct factors.
Our goal is to be able to either identify a whole number as prime or
break the number up into the product of prime powers. We call this
the prime factorization.
Prime Factorization
Example
Find the prime factorization of 24.
Prime Factorization
Example
Find the prime factorization of 24.
24
Prime Factorization
Example
Find the prime factorization of 24.
24
4
6
Prime Factorization
Example
Find the prime factorization of 24.
24
4
6
Prime Factorization
Example
Find the prime factorization of 24.
24
6
4
2
2
2
3
Prime Factorization
Example
Find the prime factorization of 24.
24
6
4
2
2
2
3
So, the prime factorization of is given by 24 = 23 · 3.
Unique Factorization?
Example
Find the prime factorization of 24 by starting with different factors.
Unique Factorization?
Example
Find the prime factorization of 24 by starting with different factors.
24
Unique Factorization?
Example
Find the prime factorization of 24 by starting with different factors.
24
3
8
Unique Factorization?
Example
Find the prime factorization of 24 by starting with different factors.
24
3
8
Unique Factorization?
Example
Find the prime factorization of 24 by starting with different factors.
24
3
8
Unique Factorization?
Example
Find the prime factorization of 24 by starting with different factors.
24
3
8
2
4
Unique Factorization?
Example
Find the prime factorization of 24 by starting with different factors.
24
3
8
2
4
Unique Factorization?
Example
Find the prime factorization of 24 by starting with different factors.
24
3
8
2
4
Unique Factorization?
Example
Find the prime factorization of 24 by starting with different factors.
24
3
8
2
4
2
2
Unique Factorization?
Example
Find the prime factorization of 24 by starting with different factors.
24
3
8
2
4
2
2
Fundamental Theorem of Arithmetic
Is this unique factorization a fluke or is this expected?
Fundamental Theorem of Arithmetic
Is this unique factorization a fluke or is this expected?
The Fundamental Theorem of Arithmetic
The prime factorization of any whole number is unique, up to the
order of the factors.
Examples
Example
Find the prime factorization of each of the following whole numbers.
100
Examples
Example
Find the prime factorization of each of the following whole numbers.
100 = 22 · 52
Examples
Example
Find the prime factorization of each of the following whole numbers.
100 = 22 · 52
Because each factor is raised to the same power, we call this a
perfect square.
Examples
Example
Find the prime factorization of each of the following whole numbers.
100 = 22 · 52
Because each factor is raised to the same power, we call this a
perfect square.
We can also think of perfect squares as having an odd number of
factors.
Examples
Example
Find the prime factorization of each of the following whole numbers.
100 = 22 · 52
Because each factor is raised to the same power, we call this a
perfect square.
We can also think of perfect squares as having an odd number of
factors.
3250
Examples
Example
Find the prime factorization of each of the following whole numbers.
100 = 22 · 52
Because each factor is raised to the same power, we call this a
perfect square.
We can also think of perfect squares as having an odd number of
factors.
3250 = 2 · 53 · 13
Examples
Example
Find the prime factorization of each of the following whole numbers.
100 = 22 · 52
Because each factor is raised to the same power, we call this a
perfect square.
We can also think of perfect squares as having an odd number of
factors.
3250 = 2 · 53 · 13
6468
Examples
Example
Find the prime factorization of each of the following whole numbers.
100 = 22 · 52
Because each factor is raised to the same power, we call this a
perfect square.
We can also think of perfect squares as having an odd number of
factors.
3250 = 2 · 53 · 13
6468 = 22 · 3 · 72 · 11
Examples
Example
Find the prime factorization of 14053.
Examples
Example
Find the prime factorization of 14053.
Where do we start?
Examples
Example
Find the prime factorization of 14053.
Where do we start? Since none of our tests work, how far do we have
to go with checking primes?
Examples
Example
Find the prime factorization of 14053.
Where do we start? Since none of our tests work, how far do we have
to go with checking primes?
√
14053 ≈ 118.5 ⇒ 113
Examples
Example
Find the prime factorization of 14053.
Where do we start? Since none of our tests work, how far do we have
to go with checking primes?
√
14053 ≈ 118.5 ⇒ 113
14053 = 13 · 1081
Examples
Example
Find the prime factorization of 14053.
Where do we start? Since none of our tests work, how far do we have
to go with checking primes?
√
14053 ≈ 118.5 ⇒ 113
14053 = 13 · 1081
Now how far do we need to check?
Examples
Example
Find the prime factorization of 14053.
Where do we start? Since none of our tests work, how far do we have
to go with checking primes?
√
14053 ≈ 118.5 ⇒ 113
14053 = 13 · 1081
Now how far do we need to check?
√
1081 ≈ 32.87 → 31
Examples
Example
Find the prime factorization of 14053.
Where do we start? Since none of our tests work, how far do we have
to go with checking primes?
√
14053 ≈ 118.5 ⇒ 113
14053 = 13 · 1081
Now how far do we need to check?
√
1081 ≈ 32.87 → 31
14053 = 13 · 23 · 47
Factorization Theorems
Theorem
If d is a divisor of n, then
n
d
is a divisor of n.
Factorization Theorems
Theorem
If d is a divisor of n, then
n
d
is a divisor of n.
Theorem
If n is composite, then n has a prime factor p such that p2 ≤ n.
Factorization Theorems
Theorem
If d is a divisor of n, then
n
d
is a divisor of n.
Theorem
If n is composite, then n has a prime factor p such that p2 ≤ n.
Theorem
If n is a whole number greater than 1 and is not divisible by any prime
such that p2 ≤ n, then n is prime.
How Many Primes?
How many prime numbers are there?
How Many Primes?
How many prime numbers are there?
We think there are an infinite number, but ...
How Many Primes?
How many prime numbers are there?
We think there are an infinite number, but ...
If p1 , p2 , . . . , pk are a finite list of all primes, then what is
p1 · p2 · . . . · pk + 1
How Many Factors?
Example
List all of the factors of 24.
How Many Factors?
Example
List all of the factors of 24.
The factors are {1, 2, 3, 4, 6, 8, 12, 24}.
How Many Factors?
Example
List all of the factors of 24.
The factors are {1, 2, 3, 4, 6, 8, 12, 24}.
Do you notice anything about the factorizations of the factors in terms
of 2 and 3?
How Many Factors?
Example
List all of the factors of 24.
The factors are {1, 2, 3, 4, 6, 8, 12, 24}.
Do you notice anything about the factorizations of the factors in terms
of 2 and 3?
1 = 20 · 30
How Many Factors?
Example
List all of the factors of 24.
The factors are {1, 2, 3, 4, 6, 8, 12, 24}.
Do you notice anything about the factorizations of the factors in terms
of 2 and 3?
1 = 20 · 30
2 = 21 · 30
How Many Factors?
Example
List all of the factors of 24.
The factors are {1, 2, 3, 4, 6, 8, 12, 24}.
Do you notice anything about the factorizations of the factors in terms
of 2 and 3?
1 = 20 · 30
2 = 21 · 30
3 = 20 · 31
How Many Factors?
Example
List all of the factors of 24.
The factors are {1, 2, 3, 4, 6, 8, 12, 24}.
Do you notice anything about the factorizations of the factors in terms
of 2 and 3?
1 = 20 · 30
2 = 21 · 30
3 = 20 · 31
4 = 22 · 30
How Many Factors?
Example
List all of the factors of 24.
The factors are {1, 2, 3, 4, 6, 8, 12, 24}.
Do you notice anything about the factorizations of the factors in terms
of 2 and 3?
1 = 20 · 30
2 = 21 · 30
3 = 20 · 31
4 = 22 · 30
6 = 21 · 31
How Many Factors?
Example
List all of the factors of 24.
The factors are {1, 2, 3, 4, 6, 8, 12, 24}.
Do you notice anything about the factorizations of the factors in terms
of 2 and 3?
1 = 20 · 30
2 = 21 · 30
3 = 20 · 31
4 = 22 · 30
6 = 21 · 31
8 = 23 · 30
How Many Factors?
Example
List all of the factors of 24.
The factors are {1, 2, 3, 4, 6, 8, 12, 24}.
Do you notice anything about the factorizations of the factors in terms
of 2 and 3?
1 = 20 · 30
2 = 21 · 30
3 = 20 · 31
4 = 22 · 30
6 = 21 · 31
8 = 23 · 30
12 = 22 · 31
How Many Factors?
Example
List all of the factors of 24.
The factors are {1, 2, 3, 4, 6, 8, 12, 24}.
Do you notice anything about the factorizations of the factors in terms
of 2 and 3?
1 = 20 · 30
2 = 21 · 30
3 = 20 · 31
4 = 22 · 30
6 = 21 · 31
8 = 23 · 30
12 = 22 · 31
24 = 23 · 31
How Many Factors?
Example
How many factors does 36 have?
How Many Factors?
Example
How many factors does 36 have?
The prime factorization of 36 is
How Many Factors?
Example
How many factors does 36 have?
The prime factorization of 36 is 22 · 32 .
How Many Factors?
Example
How many factors does 36 have?
The prime factorization of 36 is 22 · 32 .
36 has how many factors?
How Many Factors?
Example
How many factors does 36 have?
The prime factorization of 36 is 22 · 32 .
36 has how many factors? 9
How Many Factors?
Example
How many factors does 36 have?
The prime factorization of 36 is 22 · 32 .
36 has how many factors? 9
Pattern?
How Many Factors?
Example
How many factors does 36 have?
The prime factorization of 36 is 22 · 32 .
36 has how many factors? 9
Pattern? Pattern based on factorization?
How Many Factors?
Example
How many factors does 36 have?
The prime factorization of 36 is 22 · 32 .
36 has how many factors? 9
Pattern? Pattern based on factorization?
Number of Factors
If n = pn11 · pn22 · . . . · pnk k , then n has (n1 + 1)(n2 + 1) · · · (nk + 1)
distinct factors.
Listing Factors from Factorization
Example
List all of the factors of 45.
Listing Factors from Factorization
Example
List all of the factors of 45.
Product of factors
Factor
Listing Factors from Factorization
Example
List all of the factors of 45.
Product of factors
30 · 50
Factor
1
Listing Factors from Factorization
Example
List all of the factors of 45.
Product of factors
30 · 50
31 · 50
Factor
1
3
Listing Factors from Factorization
Example
List all of the factors of 45.
Product of factors
30 · 50
31 · 50
32 · 50
Factor
1
3
9
Listing Factors from Factorization
Example
List all of the factors of 45.
Product of factors
30 · 50
31 · 50
32 · 50
30 · 51
Factor
1
3
9
5
Listing Factors from Factorization
Example
List all of the factors of 45.
Product of factors
30 · 50
31 · 50
32 · 50
30 · 51
31 · 51
Factor
1
3
9
5
15
Listing Factors from Factorization
Example
List all of the factors of 45.
Product of factors
30 · 50
31 · 50
32 · 50
30 · 51
31 · 51
32 · 51
Factor
1
3
9
5
15
45
Classifications by Factors
Definition
A divisor of n is proper if it is not equal to n.
Classifications by Factors
Definition
A divisor of n is proper if it is not equal to n.
Definition
A whole number is perfect if it equals the sum of its proper factors.
Classifications by Factors
Definition
A divisor of n is proper if it is not equal to n.
Definition
A whole number is perfect if it equals the sum of its proper factors.
Example
6 is a perfect number because 1 + 2 + 3 = 6.
Classifications by Factors
Definition
A divisor of n is proper if it is not equal to n.
Definition
A whole number is perfect if it equals the sum of its proper factors.
Example
6 is a perfect number because 1 + 2 + 3 = 6.
What is the next perfect number?
Classifications by Factors
Definition
A divisor of n is proper if it is not equal to n.
Definition
A whole number is perfect if it equals the sum of its proper factors.
Example
6 is a perfect number because 1 + 2 + 3 = 6.
What is the next perfect number? 28
Classification by Factors
While searching for 28, did you find any whole numbers whose sum
of proper factors exceeds the whole number itself?
Classification by Factors
While searching for 28, did you find any whole numbers whose sum
of proper factors exceeds the whole number itself?
Definition
A whole number is abundant if it is larger than the sum of its proper
factors.
Classification by Factors
While searching for 28, did you find any whole numbers whose sum
of proper factors exceeds the whole number itself?
Definition
A whole number is abundant if it is larger than the sum of its proper
factors.
Any you found where the sum of proper factors was less than the
whole number?
Classification by Factors
While searching for 28, did you find any whole numbers whose sum
of proper factors exceeds the whole number itself?
Definition
A whole number is abundant if it is larger than the sum of its proper
factors.
Any you found where the sum of proper factors was less than the
whole number?
Definition
A whole number is deficient if it is smaller than the sum of its proper
factors.