Whole Numbers

Slide 1 / 69

· Prime and Composite Numbers

Table of Contents

· Prime Factorization· Common Factors· Greatest Common Factor· Relatively Prime· Least Common Multiple

Slide 2 / 69

Prime and Composite Numbers

Slide 3 / 69

page46svgpage2svgpage35svgpage32svgpage8svgpage52svg

Slide 4 / 69

1 The smallest prime number is _______.

Slide 5 / 69

2 49 is not a prime number.

True

False

Slide 6 / 69

3 This list contains 3 prime numbers:1, 2, 3, 5, 9, and 12True

False

Slide 7 / 69

4 This list contains 3 prime numbers:5, 9, 20, 31, 42, 53, and 63True

False

Slide 8 / 69

5 This list contains 3 prime numbers:5, 9, 20, 31, 42, 53, and 63True

False

Slide 9 / 69

6 This list contains 3 prime numbers:15, 19, 23, 37, 47, 55, and 63True

False

Slide 10 / 69

7 This list contains 3 prime numbers:25, 29, 33, 38, 45, 57, and 76True

False

Slide 11 / 69

The Sieve of ErastosenesFind the prime numbers by sifting out the multiples of each prime.

Example:

2 is prime.

Multiples of 2: 2, 4, 6, 8, 10, 12, 14...

How do we know that the multiples of 2 are not prime?

Slide 12 / 69

The Sieve of ErastosenesSift out the multiplesof each prime.

What areyou left with?

Slide 13 / 69

A Composite Number can be divided evenly by numbers other than 1 or itself.

Examples:

1 is NOT composite. Why not?

X

Slide 14 / 69

Is 18 prime or composite? Explain

18 is composite because it can be divided evenly by more than 1 and itself. 18 can be evenly divided by: 1, 2, 3, 6, 9, and 18.

Is 63 prime or composite? Explain

63 is composite because it can be divided evenly by more than 1 and itself. 63 can be evenly divided by: 1, 3, 7, 9, 21, and 63.

Slide 15 / 69

Slide 16 / 69

8 43 is _________

A Prime

B Composite

Slide 17 / 69

9 30 is _________

A Prime

B Composite

Slide 18 / 69

10 33 is _________

A Prime

B Composite

Slide 19 / 69

Factoring a Number

Slide 20 / 69

Factors are the numbers you multiply together to get another number.

Example: 3 and 6 are factors of 18, because 3 x 6 = 18. Also, 2 x 9 =18, so 2 and 9 are also factors of 18.What are two other factors of 18?

Factors

Slide 21 / 69

Prime Factorization

is the process of factoring a number so that all of the factors are prime numbers.

Slide 22 / 69

Process for factoring a number into primes

1. Divide the given number by the smallest prime number possible.

2. Continue to divide by the smallest prime number possible.

3. Keep dividing until the quotient (answer) is one.

2 12

3

6

3

2

Example:

1

12 = 2 x 2 x 3

= 22 x 3

Slide 23 / 69

What is the prime factorization of 18?

2

3

3

18

9

31

18 = 2 x 3 x 3

= 2 x 32click for answer

Slide 24 / 69

What is the prime factorization of 24?

2

2

2

24

12

6

1

24 = 2 x 2 x 2 x 3

= 23 x 3

33

click for answer

Slide 25 / 69

11 What is the prime factorization of 30?

A 2 x 3 x 5

B 6 x 5

C 5 x 6

D 2 x 15

Slide 26 / 69

12 What is the prime factorization of 24?

A 3 x 8

B 2 x 2 x 6

C 23 x 3

D 2 x 2 x 2 x 3

Slide 27 / 69

13 What is the prime factorization of 45?

A 3 x 15

B 32 x 5

C 9 x 5

D 52 x 3

Slide 28 / 69

14 What is the prime factorization of 60?

A 2 x 3 x 10

B 2 x 5 x 2 x 3

C 22 x 3 x 5

D 22 x 15

Slide 29 / 69

15 What is the prime factorization of 100?

A 2 x 3 x 10

B 2 x 5 x 2 x 3

C 22 x 3 x 5

D 22 x 15

Slide 30 / 69

Common FactorsA common factor is a number that is a factor of two or more numbers.Find the common factors of 12 and 16.

Factors of 12: 1, 2, 3, 4, 6, 12

Factors of 16: 1, 2, 4, 8, 16

Common factors: 1, 2, 4

What is the Greatest Common Factor?

Greatest Common Factor: 4

click for answer

click for answer

click for answer

click for answer

Slide 31 / 69

Common FactorsFind the common factors of 18 and 24.

Factors of 18: 1, 2, 3, 6, 9, 18

Factors of 24: 1, 2, 3, 4, 6, 8,12, 24

Common factors: 1, 2, 3, 4, 6

What is the Greatest Common Factor?

Greatest Common Factor: 6

click for answer

click for answer

click for answer

click for answer

Slide 32 / 69

16 The greatest common factor for 12 and 48 is ____.

A 2

B 4

C 6

D 12

Slide 33 / 69

17 The greatest common factor for 24 and 36 is ____.

A 2

B 4

C 6

D 12

Slide 34 / 69

18 The greatest common factor for 42 and 64 is ____.

A 2

B 4

C 6

D 8

Slide 35 / 69

19 The greatest common factor for 50 and 100 is ____.

A 5

B 10

C 25

D 50

Slide 36 / 69

20 The greatest common factor for 36 and 90 is ____.

A 3

B 9

C 12

D 18

Slide 37 / 69

We can use prime factorization to find the greatest common factor (GCF). 1. Factor the given numbers into primes.

2. Circle the factors that are common.

3. Multiply the common factors together to find the greatest common factor.

Greatest Common Factor

Slide 38 / 69

Slide 39 / 69

Slide 40 / 69

Slide 41 / 69

21 Use prime factorization to find the GCF of 18 and 44.

Slide 42 / 69

22 Use prime factorization to find the GCF of 28 and 70.

Slide 43 / 69

23 Use prime factorization to find the GCF of 55 and 110.

Slide 44 / 69

24 Use prime factorization to find the GCF of 52 and 78.

Slide 45 / 69

25 Use prime factorization to find the GCF of 72 and 75.

Slide 46 / 69

Relatively Prime: Two or more numbers are relatively prime if their greatest common factor is 1.

Example:15 and 32 are relatively prime because their GCF is 1.

Name two numbers that are relatively prime.

Slide 47 / 69

26 Identify at least two numbers that are relatively prime to 9.

A 16

B 15

C 28

D 36

Slide 48 / 69

27 7 and 35 are not relatively prime.

True

False

Slide 49 / 69

28 Name a number that is relatively prime to 20.

Slide 50 / 69

29 Name a number that is relatively prime to 5 and 18.

Slide 51 / 69

30 Find two numbers that are relatively prime

A 7

B 14

C 15

D 49

Slide 52 / 69

Least Common Multiple

Slide 53 / 69

A multiple of a whole number is the product of the number and any nonzero whole number.

A multiple that is shared by two or more numbers is a common multiple.

Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, ...

Multiples of 14: 14, 28, 42, 56, 70, 84,...

The least of the common multiples of two or more numbers is the least common multiple (LCM). The LCM of 6 and 14 is 42.

Slide 54 / 69

Find the least common multiple of 18 and 24.

Multiples of 18: 18, 36, 54, 72, ...

Multiples of 24: 24, 48, 72, ...

LCM: 72

Slide 55 / 69

31 Find the least common multiple of 10 and 14.

A 2

B 20

C 70

D 140

Slide 56 / 69

32 Find the least common multiple of 5 and 30.

A 6

B 10

C 30

D 150

Slide 57 / 69

33 Find the least common multiple of 9 and 15.

A 3

B 30

C 45

D 135

Slide 58 / 69

34 Find the least common multiple of 3, 6, and 9.

A 3

B 12

C 18

D 36

Slide 59 / 69

35 Find the least common multiple of 16, 20, and 30.

A 80

B 100

C 240

D 320

Slide 60 / 69

Another way to find the least common multiple (LCM) is to factor the numbers into primes and then multiply all of the factors, using each common factor only once.

2 12

3

62

31

Example: Find the LCM of 12 and 18.

2 18

3

3

9

31

12 = 2 x 2 x 3

18 = 2 x 3 x 3

LCM: 2 x 3 x 2 x 3 = 36

Slide 61 / 69

Find the least common multiple (LCM) by factoring the number into primes and then multiply all of the factors, using each common factor only once.

2 16

2

82

Example: Find the LCM of 16 and 28.

2 28

2

7

14

71

16 = 2 x 2 x 2 x 2

28 = 2 x 2 x 7

LCM: 2 x 2 x 2 x 2 x 7 = 112

1

42 2

Slide 62 / 69

2 20

102

5 51

Example: Find the LCM of 10, 12, and 20.

2 10

5 5

10 = 2 x 5

12 = 2 x 2 x 3

20 = 2 x 2 x 5

LCM: 2 x 5 x 2 x 3 x 5 = 300

Find the least common multiple (LCM) by factoring the numbers into primes and then multiply all of the factors, using each common factor only once.

2 12

62

3 3

11

Slide 63 / 69

36 Use prime factorization to find the LCM of 12 and 20.

Slide 64 / 69

37 Use prime factorization to find the LCM of 24 and 60.

Slide 65 / 69

38 Use prime factorization to find the LCM of 9, 15, and 18.

Slide 66 / 69

39 Use prime factorization to find the LCM of 16, 24, and 32.

Slide 67 / 69

40 Use prime factorization to find the LCM of 15, 20, 75.

Slide 68 / 69

41 Use prime factorization to find the GCF of 15, 20, 75.

Slide 69 / 69