§ 4-2 Prime and Composite Numbers

Definitions

What makes a whole number prime?

DefinitionA prime number is a whole number that has exactly two distinctfactors, one and itself.

If a number is not prime, then it is ...

DefinitionA composite number is a whole number that has more than twodistinct factors.

Our goal is to be able to either identify a whole number as prime orbreak the number up into the product of prime powers. We call thisthe prime factorization.

Definitions

What makes a whole number prime?

DefinitionA prime number is a whole number that has exactly two distinctfactors, one and itself.

If a number is not prime, then it is ...

DefinitionA composite number is a whole number that has more than twodistinct factors.

Our goal is to be able to either identify a whole number as prime orbreak the number up into the product of prime powers. We call thisthe prime factorization.

Definitions

What makes a whole number prime?

DefinitionA prime number is a whole number that has exactly two distinctfactors, one and itself.

If a number is not prime, then it is ...

DefinitionA composite number is a whole number that has more than twodistinct factors.

Our goal is to be able to either identify a whole number as prime orbreak the number up into the product of prime powers. We call thisthe prime factorization.

Definitions

What makes a whole number prime?

DefinitionA prime number is a whole number that has exactly two distinctfactors, one and itself.

If a number is not prime, then it is ...

DefinitionA composite number is a whole number that has more than twodistinct factors.

Our goal is to be able to either identify a whole number as prime orbreak the number up into the product of prime powers. We call thisthe prime factorization.

Prime Factorization

ExampleFind the prime factorization of 24.

24

4 6

322 2 322 2

So, the prime factorization of is given by 24 = 23 · 3.

Prime Factorization

ExampleFind the prime factorization of 24.

24

4 6

322 2 322 2

So, the prime factorization of is given by 24 = 23 · 3.

Prime Factorization

ExampleFind the prime factorization of 24.

24

4 6

322 2 322 2

So, the prime factorization of is given by 24 = 23 · 3.

Prime Factorization

ExampleFind the prime factorization of 24.

24

4 6

322 2 322 2

So, the prime factorization of is given by 24 = 23 · 3.

Prime Factorization

ExampleFind the prime factorization of 24.

24

4 6

322 2

322 2

So, the prime factorization of is given by 24 = 23 · 3.

Prime Factorization

ExampleFind the prime factorization of 24.

24

4 6

322 2 322 2

So, the prime factorization of is given by 24 = 23 · 3.

Unique Factorization?

ExampleFind the prime factorization of 24 by starting with different factors.

24

3 83

422

2 22 2

Unique Factorization?

ExampleFind the prime factorization of 24 by starting with different factors.

24

3 83

422

2 22 2

Unique Factorization?

ExampleFind the prime factorization of 24 by starting with different factors.

24

3 8

3

422

2 22 2

Unique Factorization?

ExampleFind the prime factorization of 24 by starting with different factors.

24

3 83

422

2 22 2

Unique Factorization?

ExampleFind the prime factorization of 24 by starting with different factors.

24

3 83

422

2 22 2

Unique Factorization?

ExampleFind the prime factorization of 24 by starting with different factors.

24

3 83

42

2

2 22 2

Unique Factorization?

ExampleFind the prime factorization of 24 by starting with different factors.

24

3 83

422

2 22 2

Unique Factorization?

ExampleFind the prime factorization of 24 by starting with different factors.

24

3 83

422

2 22 2

Unique Factorization?

ExampleFind the prime factorization of 24 by starting with different factors.

24

3 83

422

2 2

2 2

Unique Factorization?

ExampleFind the prime factorization of 24 by starting with different factors.

24

3 83

422

2 22 2

Fundamental Theorem of Arithmetic

Is this unique factorization a fluke or is this expected?

The Fundamental Theorem of ArithmeticThe prime factorization of any whole number is unique, up to theorder of the factors.

Fundamental Theorem of Arithmetic

Is this unique factorization a fluke or is this expected?

The Fundamental Theorem of ArithmeticThe prime factorization of any whole number is unique, up to theorder of the factors.

Examples

ExampleFind 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 aperfect square.We can also think of perfect squares as having an odd number offactors.

3250 = 2 · 53 · 13

6468 = 22 · 3 · 72 · 11

Examples

ExampleFind 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 aperfect square.We can also think of perfect squares as having an odd number offactors.

3250 = 2 · 53 · 13

6468 = 22 · 3 · 72 · 11

Examples

ExampleFind 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 aperfect square.

We can also think of perfect squares as having an odd number offactors.

3250 = 2 · 53 · 13

6468 = 22 · 3 · 72 · 11

Examples

ExampleFind 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 aperfect square.We can also think of perfect squares as having an odd number offactors.

3250 = 2 · 53 · 13

6468 = 22 · 3 · 72 · 11

Examples

ExampleFind 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 aperfect square.We can also think of perfect squares as having an odd number offactors.

3250

= 2 · 53 · 13

6468 = 22 · 3 · 72 · 11

Examples

ExampleFind 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 aperfect square.We can also think of perfect squares as having an odd number offactors.

3250 = 2 · 53 · 13

6468 = 22 · 3 · 72 · 11

Examples

ExampleFind 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 aperfect square.We can also think of perfect squares as having an odd number offactors.

3250 = 2 · 53 · 13

6468

= 22 · 3 · 72 · 11

Examples

ExampleFind 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 aperfect square.We can also think of perfect squares as having an odd number offactors.

3250 = 2 · 53 · 13

6468 = 22 · 3 · 72 · 11

Examples

ExampleFind the prime factorization of 14053.

Where do we start? Since none of our tests work, how far do we haveto 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

Examples

ExampleFind the prime factorization of 14053.

Where do we start?

Since none of our tests work, how far do we haveto 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

Examples

ExampleFind the prime factorization of 14053.

Where do we start? Since none of our tests work, how far do we haveto 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

Examples

ExampleFind the prime factorization of 14053.

Where do we start? Since none of our tests work, how far do we haveto 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

Examples

ExampleFind the prime factorization of 14053.

Where do we start? Since none of our tests work, how far do we haveto 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

Examples

ExampleFind the prime factorization of 14053.

Where do we start? Since none of our tests work, how far do we haveto 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

Examples

ExampleFind the prime factorization of 14053.

Where do we start? Since none of our tests work, how far do we haveto 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

Examples

ExampleFind the prime factorization of 14053.

Where do we start? Since none of our tests work, how far do we haveto 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

TheoremIf 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.

TheoremIf n is a whole number greater than 1 and is not divisible by any primesuch that p2 ≤ n, then n is prime.

Factorization Theorems

TheoremIf 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.

TheoremIf n is a whole number greater than 1 and is not divisible by any primesuch that p2 ≤ n, then n is prime.

Factorization Theorems

TheoremIf 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.

TheoremIf n is a whole number greater than 1 and is not divisible by any primesuch that p2 ≤ n, then n is prime.

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 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 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?

ExampleList 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 termsof 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?

ExampleList 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 termsof 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?

ExampleList 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 termsof 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?

ExampleList 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 termsof 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?

ExampleList 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 termsof 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?

ExampleList 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 termsof 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?

ExampleList 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 termsof 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?

ExampleList 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 termsof 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?

ExampleList 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 termsof 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?

ExampleList 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 termsof 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?

ExampleList 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 termsof 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?

ExampleHow 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 · p

n22 · . . . · p

nkk , then n has (n1 + 1)(n2 + 1) · · · (nk + 1)

distinct factors.

How Many Factors?

ExampleHow 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 · p

n22 · . . . · p

nkk , then n has (n1 + 1)(n2 + 1) · · · (nk + 1)

distinct factors.

How Many Factors?

ExampleHow 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 · p

n22 · . . . · p

nkk , then n has (n1 + 1)(n2 + 1) · · · (nk + 1)

distinct factors.

How Many Factors?

ExampleHow 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 · p

n22 · . . . · p

nkk , then n has (n1 + 1)(n2 + 1) · · · (nk + 1)

distinct factors.

How Many Factors?

ExampleHow 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 · p

n22 · . . . · p

nkk , then n has (n1 + 1)(n2 + 1) · · · (nk + 1)

distinct factors.

How Many Factors?

ExampleHow 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 · p

n22 · . . . · p

nkk , then n has (n1 + 1)(n2 + 1) · · · (nk + 1)

distinct factors.

How Many Factors?

ExampleHow 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 · p

n22 · . . . · p

nkk , then n has (n1 + 1)(n2 + 1) · · · (nk + 1)

distinct factors.

How Many Factors?

ExampleHow 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 · p

n22 · . . . · p

nkk , then n has (n1 + 1)(n2 + 1) · · · (nk + 1)

distinct factors.

Listing Factors from Factorization

ExampleList all of the factors of 45.

Product of factors Factor30 · 50 131 · 50 332 · 50 930 · 51 531 · 51 1532 · 51 45

Listing Factors from Factorization

ExampleList all of the factors of 45.

Product of factors Factor

30 · 50 131 · 50 332 · 50 930 · 51 531 · 51 1532 · 51 45

Listing Factors from Factorization

ExampleList all of the factors of 45.

Product of factors Factor30 · 50 1

31 · 50 332 · 50 930 · 51 531 · 51 1532 · 51 45

Listing Factors from Factorization

ExampleList all of the factors of 45.

Product of factors Factor30 · 50 131 · 50 3

32 · 50 930 · 51 531 · 51 1532 · 51 45

Listing Factors from Factorization

ExampleList all of the factors of 45.

Product of factors Factor30 · 50 131 · 50 332 · 50 9

30 · 51 531 · 51 1532 · 51 45

Listing Factors from Factorization

ExampleList all of the factors of 45.

Product of factors Factor30 · 50 131 · 50 332 · 50 930 · 51 5

31 · 51 1532 · 51 45

Listing Factors from Factorization

ExampleList all of the factors of 45.

Product of factors Factor30 · 50 131 · 50 332 · 50 930 · 51 531 · 51 15

32 · 51 45

Listing Factors from Factorization

ExampleList all of the factors of 45.

Product of factors Factor30 · 50 131 · 50 332 · 50 930 · 51 531 · 51 1532 · 51 45

Classifications by Factors

DefinitionA divisor of n is proper if it is not equal to n.

DefinitionA whole number is perfect if it equals the sum of its proper factors.

Example6 is a perfect number because 1 + 2 + 3 = 6.

What is the next perfect number? 28

Classifications by Factors

DefinitionA divisor of n is proper if it is not equal to n.

DefinitionA whole number is perfect if it equals the sum of its proper factors.

Example6 is a perfect number because 1 + 2 + 3 = 6.

What is the next perfect number? 28

Classifications by Factors

DefinitionA divisor of n is proper if it is not equal to n.

DefinitionA whole number is perfect if it equals the sum of its proper factors.

Example6 is a perfect number because 1 + 2 + 3 = 6.

What is the next perfect number? 28

Classifications by Factors

DefinitionA divisor of n is proper if it is not equal to n.

DefinitionA whole number is perfect if it equals the sum of its proper factors.

Example6 is a perfect number because 1 + 2 + 3 = 6.

What is the next perfect number?

28

Classifications by Factors

DefinitionA divisor of n is proper if it is not equal to n.

DefinitionA whole number is perfect if it equals the sum of its proper factors.

Example6 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 sumof proper factors exceeds the whole number itself?

DefinitionA whole number is abundant if it is larger than the sum of its properfactors.

Any you found where the sum of proper factors was less than thewhole number?

DefinitionA whole number is deficient if it is smaller than the sum of its properfactors.

Classification by Factors

While searching for 28, did you find any whole numbers whose sumof proper factors exceeds the whole number itself?

DefinitionA whole number is abundant if it is larger than the sum of its properfactors.

Any you found where the sum of proper factors was less than thewhole number?

DefinitionA whole number is deficient if it is smaller than the sum of its properfactors.

Classification by Factors

While searching for 28, did you find any whole numbers whose sumof proper factors exceeds the whole number itself?

DefinitionA whole number is abundant if it is larger than the sum of its properfactors.

Any you found where the sum of proper factors was less than thewhole number?

DefinitionA whole number is deficient if it is smaller than the sum of its properfactors.

Classification by Factors

While searching for 28, did you find any whole numbers whose sumof proper factors exceeds the whole number itself?

DefinitionA whole number is abundant if it is larger than the sum of its properfactors.

Any you found where the sum of proper factors was less than thewhole number?

DefinitionA whole number is deficient if it is smaller than the sum of its properfactors.