Whole Numbers Slide 1 / 69 -...
69
Slide 1 / 69 Whole Numbers
Transcript of Whole Numbers Slide 1 / 69 -...
Slide 1 / 69
Whole Numbers
Slide 2 / 69
· Prime and Composite Numbers
Table of Contents
· Prime Factorization· Common Factors· Greatest Common Factor· Relatively Prime· Least Common Multiple
Slide 3 / 69
Prime and Composite Numbers
Slide 4 / 69
A prime number can only be divided evenly by itself and one.
Examples:
One is not prime because it can ONLY be divided evenly by one.
X
Slide 5 / 69
1 The smallest prime number is _______.
Slide 6 / 69
2 49 is not a prime number.
True
False
Slide 7 / 69
3 This list contains 3 prime numbers:1, 2, 3, 5, 9, and 12True
False
Slide 8 / 69
4 This list contains 3 prime numbers:5, 9, 20, 31, 42, 53, and 63True
False
Slide 9 / 69
5 This list contains 3 prime numbers:5, 9, 20, 31, 42, 53, and 63True
False
Slide 10 / 69
6 This list contains 3 prime numbers:15, 19, 23, 37, 47, 55, and 63True
False
Slide 11 / 69
7 This list contains 3 prime numbers:25, 29, 33, 38, 45, 57, and 76True
False
Slide 12 / 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 13 / 69
The Sieve of ErastosenesSift out the multiplesof each prime.
What areyou left with?
Slide 14 / 69
A Composite Number can be divided evenly by numbers other than 1 or itself.
Examples:
1 is NOT composite. Why not?
X
Slide 15 / 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 16 / 69
Slide 17 / 69
8 43 is _________
A Prime
B Composite
Slide 18 / 69
9 30 is _________
A Prime
B Composite
Slide 19 / 69
10 33 is _________
A Prime
B Composite
Slide 20 / 69
Factoring a Number
Slide 21 / 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 22 / 69
Prime Factorization
is the process of factoring a number so that all of the factors are prime numbers.
Slide 23 / 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 24 / 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 25 / 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 26 / 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 27 / 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 28 / 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 29 / 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 30 / 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 31 / 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 32 / 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 33 / 69
16 The greatest common factor for 12 and 48 is ____.
A 2
B 4
C 6
D 12
Slide 34 / 69
17 The greatest common factor for 24 and 36 is ____.
A 2
B 4
C 6
D 12
Slide 35 / 69
18 The greatest common factor for 42 and 64 is ____.
A 2
B 4
C 6
D 8
Slide 36 / 69
19 The greatest common factor for 50 and 100 is ____.
A 5
B 10
C 25
D 50
Slide 37 / 69
20 The greatest common factor for 36 and 90 is ____.
A 3
B 9
C 12
D 18
Slide 38 / 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 39 / 69
PullPull
2
2
2
16
8
4
221
31
6
3
2
2
12
12 = 2 x 2 x 3 16 = 2 x 2 x 2 x 2
The Greatest Common Factor is 2 x 2 = 4
Use prime factorization to find the greatest common factor of 12 and 16.
Slide 40 / 69
2
2
3
36
18
9
33
1
2
3
3
90
45
15
551
Use prime factorization to find the greatest common factor of 36 and 90.
36 = 2 x 2 x 3 x 3 90 = 2 x 3 x 3 x 5
1. Factor the given number into primes.2. Circle factors that are common.3. Multiply the common factors together to find the greatest common factor.
PullPull
GCF is 2 x 3 x 3 = 18
Slide 41 / 69
2
2
3
60
30
15
55
1
2 72
2
2
36
18
93
Use prime factorization to find the greatest common factor of 60 and 72.
60 = 2 x 2 x 3 x 5
1. Factor the given number into primes.2. Circle factors that are common.3. Multiply the common factors together to find the greatest common factor.
PullPull
GCF is 2 x 2 x 3 = 12
1
33
72 = 2 x 2 x 2 x 3 x 3
Slide 42 / 69
21 Use prime factorization to find the GCF of 18 and 44.
Slide 43 / 69
22 Use prime factorization to find the GCF of 28 and 70.
Slide 44 / 69
23 Use prime factorization to find the GCF of 55 and 110.
Slide 45 / 69
24 Use prime factorization to find the GCF of 52 and 78.
Slide 46 / 69
25 Use prime factorization to find the GCF of 72 and 75.
Slide 47 / 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 48 / 69
26 Identify at least two numbers that are relatively prime to 9.
A 16
B 15
C 28
D 36
Slide 49 / 69
27 7 and 35 are not relatively prime.
True
False
Slide 50 / 69
28 Name a number that is relatively prime to 20.
Slide 51 / 69
29 Name a number that is relatively prime to 5 and 18.
Slide 52 / 69
30 Find two numbers that are relatively prime
A 7
B 14
C 15
D 49
Slide 53 / 69
Least Common Multiple
Slide 54 / 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 55 / 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 56 / 69
31 Find the least common multiple of 10 and 14.
A 2
B 20
C 70
D 140
Slide 57 / 69
32 Find the least common multiple of 5 and 30.
A 6
B 10
C 30
D 150
Slide 58 / 69
33 Find the least common multiple of 9 and 15.
A 3
B 30
C 45
D 135
Slide 59 / 69
34 Find the least common multiple of 3, 6, and 9.
A 3
B 12
C 18
D 36
Slide 60 / 69
35 Find the least common multiple of 16, 20, and 30.
A 80
B 100
C 240
D 320
Slide 61 / 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 62 / 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 63 / 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 64 / 69
36 Use prime factorization to find the LCM of 12 and 20.
Slide 65 / 69
37 Use prime factorization to find the LCM of 24 and 60.
Slide 66 / 69
38 Use prime factorization to find the LCM of 9, 15, and 18.
Slide 67 / 69
39 Use prime factorization to find the LCM of 16, 24, and 32.
Slide 68 / 69
40 Use prime factorization to find the LCM of 15, 20, 75.
Slide 69 / 69
41 Use prime factorization to find the GCF of 15, 20, 75.
