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


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


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


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


Vocabulary Greatest common

factor : The number with the largest value that is a factor for more than one number (a set or pair)


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


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


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


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


Vocabulary

Prime factorization : Breaking a number down into factors that are all prime numbers


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


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


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


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


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


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


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


Guided practice 1, greatest common factor, ‘prime product’ method, continued

What is the prime factorization of 63?

What is the prime factorization of 84?


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


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


Greatest common factors of number sets

Use the same process to find common factors even if you are working with more than two numbers


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


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


Example 2, greatest common factor of a number set, ‘prime product’ method, continued

Next, prime factor 108



Then, prime factor 216



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


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



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


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



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?


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


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


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?


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


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The Number System Greatest Common Factor 1 © 2013 Meredith S. Moody