The Number System Greatest Common Factor 1 © 2013 Meredith S. Moody.
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Transcript of The Number System Greatest Common Factor 1 © 2013 Meredith S. Moody.
The Number System
Greatest Common Factor
1© 2013 Meredith S. Moody
Objective: You will be able to…
Find all the factors of a number
Prime factor a number Find the greatest common
factor for a pair or set of numbers
2© 2013 Meredith S. Moody
Vocabulary Factor: Any number that is
divisible by itself (or another whole number larger than itself) a whole number of times Any two numbers are factors of
another number when they are multiplied together and their product is that number
3© 2013 Meredith S. Moody
Example 1, factoring
The number 6 has four factors: 1, 2, 3, & 6 1 x 6 = 6 2 x 3 = 6 1, 2, 3, & 6 are factors of 6
4© 2013 Meredith S. Moody
Guided practice 1, finding factors
Find the factors of the number 10 1 x ? = 10 1 x 10 = 10 2 x ? = 10 2 x 5 = 10 3 x ? = 10 none 4 x ? = 10 none 5 x ? = 10 5 x 2 = 10 we already
have 5 & 2, so we have found all the factors of 10 1, 2, 5, and 10
5© 2013 Meredith S. Moody
Vocabulary Greatest common
factor : The number with the largest value that is a factor for more than one number (a set or pair)
6© 2013 Meredith S. Moody
Example 1, common factors, ‘list’ method
The numbers 6 and 8 have a common factor
The factors of 6 are 1, 2, 3, & 6 The factors of 8 are 1, 2, 4, & 8 The common factor of 6 and 8 is
2 We do not include ‘1’ because it is
a factor of all other numbers7© 2013 Meredith S. Moody
Example 1, greatest common factor, ‘list’ method
The numbers 28 and 44 have more than one common factor
The factors of 28 are 1, 2, 4, 7, 14, and 28
The factors of 44 are 1, 2, 4, 11, and 22
The common factors of 28 and 44 are 2 & 4
The greatest common factor of 28 & 44 is 4 8© 2013 Meredith S. Moody
Guided practice 1, greatest common factor, ‘list’ method Find the greatest common factor of the
numbers 30 and 50 What are the factors of 30?
1, 2, 3, 5, 6, 10, 15 What are the factors of 50?
1, 2, 5, 10, 25, 50 What are the numbers that are factors of
both numbers? 2, 5, & 10
What is the greatest common factor? 10
9© 2013 Meredith S. Moody
Vocabulary
Prime number: A number with ONLY 2 factors: 1 and the number itself . For example, 5 only 1 and 5 are the
factors of 5 5 is a prime number The number “1” is a special case
10© 2013 Meredith S. Moody
Example, prime numbers
The number 7 is a prime number Only 1 x 7 = 7 There are no other numbers whose
product is 7 Therefore, 7 is a prime number
11© 2013 Meredith S. Moody
Prime numbers: You try Give 3 examples of prime
numbers 2, 3, 5, 7, 11, 13, 17, 19, 23, 29…
How do you know they are prime? The only factors are 1 and itself
12© 2013 Meredith S. Moody
Vocabulary
Prime factorization : Breaking a number down into factors that are all prime numbers
13© 2013 Meredith S. Moody
Example 1, prime factorization Two factors of 12 are 3 and 4 3 x 4 = 12 3 is a prime number Is 4 a prime number? No 2 x 2 = 4; is 2 a prime number? Yes Therefore, 3 x 2 x 2 = 12 3 and 2 are prime numbers, so the prime
factorization of 12 is 3 x 2 x 2
14© 2013 Meredith S. Moody
Example 1, continued Two other factors of 12 are 2 and 6 2 x 6 = 12 2 is a prime number Is 6 a prime number? No 2 x 3 = 6 2 is prime; is 3 a prime number? Yes Therefore, 2 x 2 x 3 = 12 2 and 3 are prime numbers, so the prime
factorization of 12 is 2 x 2 x 3 no matter which factors you begin with
15© 2013 Meredith S. Moody
Vocabulary
Factor tree: A diagram used to break down a number by dividing it by its factors until all the numbers left are prime
© 2013 Meredith S. Moody 16
Example 1, factor trees
Factor trees are a useful tool when prime factoring a number
Here is a factor tree for the number 42
The prime factorization is 2 x 3 x 717© 2013 Meredith S. Moody
Example 2, factor trees Here is a longer factor tree Notice how the prime numbers are circled
as they are found
The prime factorization of 72 is 2 x 2 x 2 x 3 x 3
18© 2013 Meredith S. Moody
Factor trees: You try Make a factor tree for the number 108
and list the prime factorization
The prime factorization is 2 x 2 x 3 x 3 x 3
19© 2013 Meredith S. Moody
Guided practice 1, prime factorization
Find the prime factorization of 50 5 x 10 = 50 Is 5 a prime number? Yes Is 10 a prime number? No 2 x 5 = 10 Is 2 a prime number? Yes; is 5 a prime
number? Yes What is the prime factorization of 50?
2 x 5 x 5 = 50
20© 2013 Meredith S. Moody
Prime factorization: You try
Find the prime factorization of 32
The prime factorization of 32 is 2 x 2 x 2 x 2 x 2
21© 2013 Meredith S. Moody
Guided practice 1, greatest common factor, ‘prime product’ method
You can use the prime factorization to find the greatest common factor
Find the prime factorization of both numbers and multiply the common prime factors
Find the greatest common factor of 63 and 84 using the prime factorization method
22© 2013 Meredith S. Moody
Guided practice 1, greatest common factor, ‘prime product’ method, continued
What is the prime factorization of 63?
What is the prime factorization of 84?
© 2013 Meredith S. Moody 23
Guided practice 1, greatest common factor, ‘prime product’ method, continued
What are the common prime factors of 63 and 84?
3 and 7 What is the product of those common
prime factors? 3 x 7 = 21
What is the greatest common factor of 63 and 84?
21
24© 2013 Meredith S. Moody
Which is better? Why would you tell someone to use the
list method if they want to find the greatest common factor?
Why would you tell someone to use the prime product method if they want to find the greatest common factor?
Is there a time one is better than the other?
The list method works better with smaller numbers
The prime product works better with larger numbers
Does it matter which one you use? Technically, no – although using the
appropriate method will save you time
25© 2013 Meredith S. Moody
Greatest common factors of number sets
Use the same process to find common factors even if you are working with more than two numbers
26© 2013 Meredith S. Moody
Example 1, greatest common factor of a number set, ‘list’ method
Find the greatest common factor for the number set {15, 75, 150}
Factors of 15 are 1, 3, 5, & 15 Factors of 75 are 1, 3, 5, 15, 25, & 75 Factors of 150 are 1, 2, 3, 5, 6, 10,
15, 25, 30, 50, 75, & 150 The common factors are 3, 5, & 15 The greatest common factor is 15
27© 2013 Meredith S. Moody
Example 2, greatest common factor of a number set, ‘prime product’ method
Find the greatest common factor of the number set {84, 108, 216}
First, prime factor 84
© 2013 Meredith S. Moody 28
Example 2, greatest common factor of a number set, ‘prime product’ method, continued
Next, prime factor 108
29© 2013 Meredith S. Moody
Example 2, greatest common factor of a number set, ‘prime product’ method, continued
Then, prime factor 216
30© 2013 Meredith S. Moody
Example 2, greatest common factor of a number set, ‘prime product’ method, continued
2 x 2 x 3 x 7 = 84 2 x 2 x 3 x 3 x 3 = 108 2 x 2 x 2 x 3 x 3 x 3 = 216 All three numbers have 2 x 2 x 3 in
common; 2 x 2 x 3 = 12 The greatest common factor of the
number set {84, 108, 216} is 12
31© 2013 Meredith S. Moody
Guided practice 1, greatest common factor
Find the greatest common factor using the list method:
24 and 40 Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24 Factors of 40: 1, 2, 4, 5, 9, 10, 20, 40 Common factors: 1, 2, 4 Greatest common factor: 4
32© 2013 Meredith S. Moody
Guided practice 2, greatest common factor
Find the greatest common factor using the prime product method:
165 and 135 Common factors: 3, 5 3 x 5 = 15 Greatest common
factor: 15
© 2013 Meredith S. Moody 33
Greatest common factor: You try #1
Find the greatest common factor using any method:
18 and 30 6
16 and 24 8
12 and 72 12
34© 2013 Meredith S. Moody
Guided practice 3, greatest common factor
Bennie is catering a party. He has 60 celery sticks and 45 small tacos. He wants both kinds of food in each plate. He wants the food distributed evenly and none left over: what is the largest number of plates he can use and how many of each type of food will be on each plate?
35© 2013 Meredith S. Moody
Guided practice 3, greatest common factor, continued
The greatest number of plates is going to be the greatest common factor of 60 and 45, because that way, there will be no food left off any of the plates
Find the greatest common factor of 60 and 45 15
36© 2013 Meredith S. Moody
Guided practice 3, greatest common factor, continued
Remember, Bennie wants the food evenly distributed
If there are 15 plates, how many celery sticks will go on each plate? 60 ÷ 15 = 4; 4 celery sticks on each
plate If there are 15 plates, how many
small tacos will go on each plate? 45 ÷ 15 = 3; 3 tacos on each plate
37© 2013 Meredith S. Moody
Greatest common factor, You try #2
Jane is putting together candy bags for her Halloween party
The candy bags are sold in bundles of 15, 20, and 25 She buys 3 bags of mini Reese’s candy with 40 cups
per bag She buys 2 bags of mini Hershey’s candy with 25 bars
per bag She buys 2 bags of mini Snicker’s candy with 30 bars
per bag What is the greatest number of bags she should buy
and how much of each type of candy should she put in each bag if she wants an even distribution?
Will there be any bags left over? If so, how many? Will there be any candy left over? If so, what kind and
how many?
38© 2013 Meredith S. Moody
Answer Jane has 120 Reese’s, 50 Hershey’s, and
60 Snickers The greatest common factor of 120, 50,
and 60 is 10 She should buy the 15 bag pack and put
12 Reese’s, 5 Hershey’s, and 6 Snickers in each bag
There will be 5 bags left over There will be no candy left over
39© 2013 Meredith S. Moody