unit 4 divisibility, factors, and multiples · pdf fileunit 4 –media lesson 1 unit 4...

Unit 4 – Media Lesson

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UNIT 4 – DIVISIBILITY, FACTORS, AND MULTIPLES

INTRODUCTION

In Units 1 and 2, we decomposed numbers additively. Specifically, we found ways we could rewrite a number

as the sum of its base 10 components or as a combination of positive and negative chips. One way we found this

decomposition useful was when we subtracted with base blocks. If we needed to find 73 − 47 we might trade

one of the seven rods for ten units so we would have enough units to subtract off the 7 ones in 47. Next we will

learn about decomposing numbers multiplicatively. This means we will look at different ways to rewrite whole

numbers as products of 2 or more factors. .

The table below shows the learning objectives that are the achievement goal for this unit. Read through them

carefully now to gain initial exposure to the terms and concept names for the lesson. Refer back to the list at the

end of the lesson to see if you can perform each objective.

Learning Objective Media Examples You Try

Given a division problem, find the quotient and remainder 1 2

Determine if a number is a factor of another number 3 4

Determine if a number is divisible by another number 3 4

List all of the factors of a number 5 6

Solve applications involving GCF and LCM 7

Find the GCF by comparing lists of factors 8 9

Find the LCM by comparing lists of multiples 10 11

Verify that a number is prime 12 13

Determine if a number is prime or composite 14 15

Find the prime factorization of a number 16 17

Use the prime factorizations of numbers to find their GCF and LCM 18 19


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UNIT 4 – MEDIA LESSON

SECTION 4.1: FACTORS AND DIVISIBILITY

Problem 1 MEDIA EXAMPLE – Division with Remainders

a) Solve the following division problem by grouping the dividend in divisor size groups. Write your result

symbolically as both multiplication and division equations.

29 ÷ 6 =

Division Equation: _______________________ Multiplication Equation: ______________________

b) Solve the following division problems using a calculator. Write your result symbolically and in words.

Also, rewrite your results in multiplication form and in words.

178 ÷ 19 =

Division Equation: ___________________ Multiplication Equation: ______________________

Problem 2 YOU TRY – Division with Remainders

a) Solve the following division problem by grouping the dividend in divisor size groups. Write your result

symbolically and in words. Also, rewrite your results in multiplication form and in words.

37 ÷ 5 =

Division Equation: ___________________ Multiplication Equation: _____________________


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b) Solve the following division problems using a calculator. Write your result symbolically and in words.

Also, rewrite your results in multiplication form and in words.

112 ÷ 12 =

Division Equation: ___________________ Multiplication Equation: ______________________

Problem 3 MEDIA EXAMPLE – Factors and Divisibility

Rewrite the factor questions as divisibility questions and the divisibility questions as factor questions.

a) Is 4 a factor of 30? Equivalent divisibility question: ___________________________

Answer with justification: ________________________________

b) Is 30 divisible by 6? Equivalent factor question: _______________________________


c) Is 7 a factor of 21? Equivalent divisibility question: ___________________________


d) Is 4 divisible by 8? Equivalent factor question: _______________________________


Problem 4 YOU TRY – Factors and Divisibility

Determine whether the answers to the following questions are yes or no. Justify your answer by showing a

corresponding multiplication or division statement.

a) Is 6 a factor of 30? Equivalent divisibility question: ___________________________


b) Is 17 divisible by 4? Equivalent factor question: _______________________________



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Problem 5 MEDIA EXAMPLE – Finding All of the Factors of a Number

Method: To determine all of the factors of a whole number, we will find all the pairs of whole numbers whose

product is the number. We will check all the numbers whose square is less than the number we are trying to

factor.

Table of Perfect Squares

22 = 4 52 = 25 82 = 64 112 = 121

32 = 9 62 = 36 92 = 81 122 = 144

42 = 16 72 = 49 102 = 100 132 = 169

Directions: Find all factors of the given numbers by finding factor pairs. Use the table of perfect squares to see

what the largest number you have to check is. Write your final answer as a list of factors separated by commas.

a) 18 Largest number you have to check: _____

List of Factors: ______________________________________________________

b) 90 Largest number you have to check: _____

List of Factors: ______________________________________________________

Problem 6 YOU TRY – Finding All of the Factors of a Number

Find all factors of the given numbers by finding factor pairs. Use the table of perfect squares to see what the

largest number you have to check is. Write your final answer as a list of factors separated by commas.

84 Largest number you have to check: _____

List of Factors: ______________________________________________________


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SECTION 4.2: GREATEST COMMON FACTOR AND LEAST COMMON MULTIPLE In this section, we will use our knowledge of factors, divisibility and primes to determine factors and multiple

that two or more numbers share.

Problem 7 MEDIA EXAMPLE – Intro to Greatest Common Factor and Least Common Multiple

a) You and your friends are sending care packages to military service members overseas. Each package

will contain brownies and cookies. You have 20 brownies and 12 cookies. Every package made needs

to be identical. What is the greatest number of packages you can send that meets this requirement?

b) Judy and Dan are running around a track. Judy can run one lap in 3 minutes while it takes Dan 4

minutes. If they both start at the same time, how many minutes will it take them to meet?


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Problem 8 MEDIA EXAMPLE – Finding the GCF of Two Numbers

Definitions:

Common Factors of two numbers are factors that both numbers share.

The Greatest Common Factor (GCF) of two numbers is the largest of these common factors.

a) Find all factors of 36. Write your final answer as a list of factors separated by commas.

List of Factors 36: ______________________________________________________

b) Find all factors of 90. Write your final answer as a list of factors separated by commas.

List of Factors of 90: ______________________________________________________

c) List the common factors of 36 and 90: _______________________________________

d) Identify the Greatest Common Factor (GCF) of 36 and 90: _________


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Problem 9 YOU TRY – Finding the GCF of Two Numbers

a) Find all factors of 24. Write your final answer as a list of factors separated by commas.

List of Factors 24: ______________________________________________________

b) Find all factors of 60. Write your final answer as a list of factors separated by commas.

List of Factors of 60: ______________________________________________________

c) List the common factors of 24 and 60: _______________________________________

d) Identify the Greatest Common Factor (GCF) of 24 and 60: _________

Problem 10 MEDIA EXAMPLE – Multiples, Common Multiples, and LCM

Definitions:

Common Multiples of two numbers are multiples that both numbers share.

The Least Common Multiple (LCM) of two numbers is the least of these common multiples

a) The first six multiples of 8 are: ___________________________________

b) The first six multiples of 12 are: ___________________________________

c) Some common multiples of 8 and 12 are: _______________________________

d) The Least Common Multiple (LCM) of 8 and 12 is: _____________


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Problem 11 YOU TRY – Multiples, Common Multiples, and LCM

a) The first six multiples of 6 are: ___________________________________

b) The first six multiples of 4 are: ___________________________________

c) Some common multiples of 6 and 4 are: _______________________________

d) The Least Common Multiple (LCM) of 6 and 4 is: _____________

SECTION 4.3: PRIME AND COMPOSITE NUMBERS In this section, we will investigate the concept of prime and composite numbers and learn how to find the prime

factorization of a number.

Problem 12 MEDIA EXAMPLE – Verifying a Number is Prime

Definitions:

A prime number is a whole number greater than 1 whose factor pairs are only the number itself and one.

A composite number is a whole number greater than 1 which has at least one factor other than itself and one.

Method:

1. The smallest prime numbers are

2, 3, 5, 7, 11, 13, 17, 19

2. To determine if a number is prime or composite, we only need to check to see if the number is divisible

by the prime factors whose square is less than the number we are trying to factor

Table of Prime Perfect Squares

22 = 4 52 = 25 112 = 121 172 = 289

32 = 9 72 = 49 132 = 169 192 = 361

Directions: Verify that the following numbers are prime by checking to see if the number is divisible by any

prime numbers whose square is less than the number given.

a) 89 Largest prime you have to check: _____

b) 163 Largest prime you have to check: _____


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Problem 13 You Try – Verifying a Number is Prime

Verify that the following numbers are prime by checking to see if the number is divisible by any prime numbers

whose square is less than the number given.

109 Largest prime you have to check: _____

Problem 14 MEDIA EXAMPLE – Prime and Composite Numbers

Determine whether the numbers are prime or composite. If it is composite, show at least one factor pair of the

number besides 1 and itself. If it is prime, show the numbers you tested and the results of your division.



c) 119 Largest prime you have to check: _____


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Problem 15 YOU TRY – Prime and Composite Numbers

Determine whether the numbers are prime or composite. If it is composite, show at least one factor pair of the

number besides 1 and itself. If it is prime, show the numbers you tested and the results of your division.



Problem 16 MEDIA EXAMPLE – Prime Factorization

Find the prime factorizations for the given numbers using factor trees. Write the final result in exponential form

and factored form.

a) 12 b) 75 c) 155

Factored Form: Factored Form: Factored Form:

Exponential Form: Exponential Form: Exponential Form:


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Problem 17 YOU TRY – Prime Factorization

Find the prime factorizations for the given numbers using factor trees. Write the final result in exponential form

and factored form.

a) 18 b) 84

Factored Form: Factored Form:

Exponential Form: Exponential Form:

SECTION 4.4: PRIME FACTORIZATION, GCF, AND LCM In this section, we are going to use prime factorization to find a more streamlined approach to finding the GCF

and LCM of two numbers.

First let’s review the method we used in 4.2 to find the GCF and LCM.

A. To find the GCF of 8 and 12, we would follow the steps below.

1. Find all the factors of 8. Factors of 8: 1, 2, 4, 8

2. Find all the factors of 12. Factors of 12: 1, 2, 3, 4, 6, 12

3. The GCF of 8 and 12 is the largest factor they have in common. So the GCF is 4.

B. To find the LCM of 8 and 12, we would follow the steps below.

1. List some multiples of 8. Multiples of 8: 8, 16, 24, 32, 40, 48, …

2. List some multiples of 12. Multiples of 12: 12, 24, 36, 48, 60, …

3. The LCM of 8 and 12 is the smallest multiple they have in common. So the LCM is 24.


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Problem 18 MEDIA EXAMPLE – Prime Factorization, GCF, and LCM

1. Use the prime factorization method to determine the GCF and LCM of 8 and 12.

a) Find the prime factorizations of 8 and 12 using factor trees and write the prime factorizations in factored

form.

8 12


b) List of common prime factors: ____________________ (include repeated factors)

c) The product of the common prime factors of 8 and 12 is their GCF. Find the GCF.

GCF of 8 and 12:______________

d) The LCM of 8 and 12 is their product divided by their GCF. Find the LCM. Show all steps.

LCM of 8 and 12:______________

2. Use the prime factorization method to determine the GCF and LCM of 54 and 90.

a) Find the prime factorizations of 54 and 90 using factor trees and write the prime factorizations in

factored form.

54 90


b) List of common prime factors: ____________________ (include repeated factors)


GCF of 54 and 90:______________


LCM of 54 and 90:______________


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Problem 19 YOU TRY – Prime Factorization, GCF, and LCM

Use the prime factorization method to determine the GCF and LCM of 18 and 84.

a) In problem 17, you found the prime factorizations of 18 and 84. List them below in factored form.


b) List the common prime factors of 18 and 84: ____________________ (include repeated factors)


GCF of 18 and 84:______________


LCM of 18 and 84:______________

unit 4 divisibility, factors, and multiples · pdf fileunit 4 –media lesson 1 unit 4...

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