Factoring Whole Numbers

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Factoring Whole Numbers"

Transcription

1 2.2 Factoring Whole Numbers 2.2 OBJECTIVES 1. Find the factors of a whole number 2. Find the prime factorization for any number 3. Find the greatest common factor (GCF) of two numbers 4. Find the GCF for a group of numbers To factor a number means to write the number as a product of its whole-number factors. Example 1 Factoring a Composite Number Factor the number The order in which you write the factors does not matter, so would also be correct. Of course, is also a correct statement. However, in this section we are interested in factors other than 1 and the given number. Factor the number CHECK YOURSELF 1 Factor 35. In writing composite numbers as a product of factors, there will be a number of different possible factorizations. Example 2 Factoring a Composite Number Find three ways to factor 72. NOTE There have to be at least two different factorizations, because a composite number has factors other than 1 and itself (1) 6 12 (2) 3 24 (3) CHECK YOURSELF 2 Find three ways to factor 42. We now want to write composite numbers as a product of their prime factors. Look again at the first factored line of Example 2. The process of factoring can be continued until all the factors are prime numbers. 137

2 138 CHAPTER 2 MULTIPLYING AND DIVIDING FRACTIONS Example 3 Factoring a Composite Number NOTE This is often called a factor tree. NOTE Finding the prime factorization of a number will be important in our later work in adding fractions is still not prime, and so we continue by factoring is now written as a product of prime factors. When we write 72 as , no further factorization is possible. This is called the prime factorization of 72. Now, what if we start with the second factored line from the same example, ? Example 3 (Continued) Factoring a Composite Number Continue to factor 6 and Continue again to factor 4. Other choices for the factors of 12 are possible. As we shall see, the end result will be the same. No matter which pair of factors you start with, you will find the same prime factorization. In this case, there are three factors of 2 and two factors of 3. Because multiplication is commutative, the order in which we write the factors does not matter. CHECK YOURSELF 3 We could also write Continue the factorization. Rules and Properties: The Fundamental Theorem of Arithmetic There is exactly one prime factorization for any composite number. NOTE The prime factorization is then the product of all the prime divisors and the final quotient. The method of the previous example will always work. However, an easier method for factoring composite numbers exists. This method is particularly useful when numbers get large, in which case factoring with a number tree becomes unwieldy. Rules and Properties: Factoring by Division To find the prime factorization of a number, divide the number by a series of primes until the final quotient is a prime number.

3 FACTORING WHOLE NUMBERS SECTION Example 4 Finding Prime Factors To write 60 as a product of prime factors, divide 2 into 60 for a quotient of 30. Continue to divide by 2 again for the quotient 15. Because 2 won t divide evenly into 15, we try 3. Because the quotient 5 is prime, we are done. NOTE Do you see how the divisibility tests are used here? 60 is divisible by 2, 30 is divisible by 2, and 15 is divisible by Prime Our factors are the prime divisors and the final quotient. We have CHECK YOURSELF 4 Complete the process to find the prime factorization of ? 2 90? 45 Remember to continue until the final quotient is prime. Writing composite numbers in their completely factored form can be simplified if we use a format called continued division. Example 5 Finding Prime Factors Using Continued Division NOTE In each short division, we write the quotient below rather than above the dividend. This is just a convenience for the next division. Use the continued-division method to divide 60 by a series of prime numbers. 2B60 Primes 2B30 3B15 5 Stop when the final quotient is prime. To write the factorization of 60, we list each divisor used and the final prime quotient. In our example, we have CHECK YOURSELF 5 NOTE Again the factors of 20, other than 20 itself, are less than 20. Find the prime factorization of 234. We know that a factor or a divisor of a whole number divides that number exactly. The factors or divisors of 20 are 1, 2, 4, 5, 10, 20 Each of these numbers divides 20 exactly, that is, with no remainder. Our work in this section involves common factors or divisors. A common factor or divisor for two numbers is any factor that divides both the numbers exactly.

4 140 CHAPTER 2 MULTIPLYING AND DIVIDING FRACTIONS Example 6 Finding Common Factors Look at the numbers 20 and 30. Is there a common factor for the two numbers? First, we list the factors. Then we circle the ones that appear in both lists. Factors 20: 1, 2, 4, 5, 10, 20 30: 1, 2, 3, 5, 6, 10, 15, 30 We see that 1, 2, 5, and 10 are common factors of 20 and 30. Each of these numbers divides both 20 and 30 exactly. Our later work with fractions will require that we find the greatest common factor (GCF) of a group of numbers. Definitions: Greatest Common Factor The greatest common factor (GCF) of a group of numbers is the largest number that will divide each of the given numbers exactly. Example 6 (Continued) Finding Common Factors In the first part of Example 6, the common factors of the numbers 20 and 30 were listed as 1, 2, 5, 10 Common factors of 20 and 30 The greatest common factor of the two numbers is then 10, because 10 is the largest of the four common factors. CHECK YOURSELF 6 List the factors of 30 and 36, and then find the greatest common factor. The method of Example 6 will also work in finding the greatest common factor of a group of more than two numbers. NOTE Looking at the three lists, we see that 1, 2, 3, and 6 are common factors. Example 7 Finding the Greatest Common Factor (GCF) by Listing Factors Find the GCF of 24, 30, and 36. We list the factors of each of the three numbers. 24: 1, 2, 3, 4, 6, 8, 12, 24 30: 1, 2, 3, 5, 6, 10, 15, 30 36: 1, 2, 3, 4, 6, 9, 12, 18, 36 6 is the greatest common factor of 24, 30, and 36.

5 FACTORING WHOLE NUMBERS SECTION CHECK YOURSELF 7 Find the greatest common factor (GCF) of 16, 24, and 32. The process shown in Example 7 is very time-consuming when larger numbers are involved. A better approach to the problem of finding the GCF of a group of numbers uses the prime factorization of each number. Let s outline the process. Step by Step: Finding the Greatest Common Factor NOTE If there are no common prime factors, the GCF is 1. Step 1 Step 2 Step 3 Write the prime factorization for each of the numbers in the group. Locate the prime factors that are common to all the numbers. The greatest common factor (GCF) will be the product of all the common prime factors. Example 8 Finding the Greatest Common Factor (GCF) Find the GCF of 20 and 30. Step 1 Write the prime factorization of 20 and Step 2 Find the prime factors common to each number and 5 are the common prime factors. Step 3 Form the product of the common prime factors is the greatest common factor. CHECK YOURSELF 8 Find the GCF of 30 and 36. To find the greatest common factor of a group of more than two numbers, we use the same process. Example 9 Finding the Greatest Common Factor (GCF) Find the GCF of 24, 30, and

6 142 CHAPTER 2 MULTIPLYING AND DIVIDING FRACTIONS 2 and 3 are the prime factors common to all three numbers is the GCF. CHECK YOURSELF 9 Find the GCF of 15, 30, and 45. NOTE If two numbers, such as 15 and 28, have no common factor other than 1, they are called relatively prime. Example 10 Finding the Greatest Common Factor (GCF) Find the greatest common factor of 15 and There are no common prime factors listed. But remember that 1 is a factor of every whole number. The greatest common factor of 15 and 28 is 1. CHECK YOURSELF 10 Find the greatest common factor of 30 and 49. CHECK YOURSELF ANSWERS , 3 14, : 1, 2, 3, 5, 6, 10, 15, 30 36: 1, 2, 3, 4, 6, 9, 12, 18, 36 6 is the greatest common factor : 1, 2, 4, 8, 16 24: 1, 2, 3, 4, 6, 8, 12, 24 32: 1, 2, 4, 8, 16, 32 The GCF is The GCF is GCF is 1; 30 and 49 are relatively prime

7 Name 2.2 Exercises Section Date Find the prime factorization of each number ANSWERS

8 ANSWERS In later mathematics courses, you often will want to find factors of a number with a given sum or difference. The following exercises use this technique. 21. Find two factors of 24 with a sum of Find two factors of 15 with a difference of Find two factors of 30 with a difference of Find two factors of 28 with a sum of Find the greatest common factor (GCF) for each of the following groups of numbers and and and and and and and and and and and and

9 ANSWERS , 36, and , 45, and , 140, and , 19, and , 75, and , 72, and A natural number is said to be perfect if it is equal to the sum of its counting number divisors, except itself. (a) Show that 28 is a perfect number. (b) Identify another perfect number less than Find the smallest natural number that is divisible by all of the following: 2, 3, 4, 6, 8, Tom and Dick both work the night shift at the steel mill. Tom has every sixth night off, and Dick has every eighth night off. If they both have August 1 off, when will they both be off together again? 46. Mercury, Venus, and Earth revolve around the sun once every 3, 7, and 12 months, respectively. If the three planets are now in the same straight line, what is the smallest number of months that must pass before they line up again? 145

10 Answers B B B , , B63 3B August

Prime Factorization 0.1. Overcoming Math Anxiety

Prime Factorization 0.1. Overcoming Math Anxiety 0.1 Prime Factorization 0.1 OBJECTIVES 1. Find the factors of a natural number 2. Determine whether a number is prime, composite, or neither 3. Find the prime factorization for a number 4. Find the GCF

More information

Previously, you learned the names of the parts of a multiplication problem. 1. a. 6 2 = 12 6 and 2 are the. b. 12 is the

Previously, you learned the names of the parts of a multiplication problem. 1. a. 6 2 = 12 6 and 2 are the. b. 12 is the Tallahassee Community College 13 PRIME NUMBERS AND FACTORING (Use your math book with this lab) I. Divisors and Factors of a Number Previously, you learned the names of the parts of a multiplication problem.

More information

Greatest Common Factor and Least Common Multiple

Greatest Common Factor and Least Common Multiple Greatest Common Factor and Least Common Multiple Intro In order to understand the concepts of Greatest Common Factor (GCF) and Least Common Multiple (LCM), we need to define two key terms: Multiple: Multiples

More information

Exponents, Factors, and Fractions. Chapter 3

Exponents, Factors, and Fractions. Chapter 3 Exponents, Factors, and Fractions Chapter 3 Exponents and Order of Operations Lesson 3-1 Terms An exponent tells you how many times a number is used as a factor A base is the number that is multiplied

More information

Grade 7/8 Math Circles Fall 2012 Factors and Primes

Grade 7/8 Math Circles Fall 2012 Factors and Primes 1 University of Waterloo Faculty of Mathematics Centre for Education in Mathematics and Computing Grade 7/8 Math Circles Fall 2012 Factors and Primes Factors Definition: A factor of a number is a whole

More information

Accuplacer Arithmetic Study Guide

Accuplacer Arithmetic Study Guide Testing Center Student Success Center Accuplacer Arithmetic Study Guide I. Terms Numerator: which tells how many parts you have (the number on top) Denominator: which tells how many parts in the whole

More information

Chapter 4 Fractions and Mixed Numbers

Chapter 4 Fractions and Mixed Numbers Chapter 4 Fractions and Mixed Numbers 4.1 Introduction to Fractions and Mixed Numbers Parts of a Fraction Whole numbers are used to count whole things. To refer to a part of a whole, fractions are used.

More information

18. [Multiples / Factors / Primes]

18. [Multiples / Factors / Primes] 18. [Multiples / Factors / Primes] Skill 18.1 Finding the multiples of a number. Count by the number i.e. add the number to itself continuously. OR Multiply the number by 1, then 2,,, 5, etc. to get the

More information

NAME TEST DATE FRACTION STUDY GUIDE/EXTRA PRACTICE PART 1: PRIME OR COMPOSITE?

NAME TEST DATE FRACTION STUDY GUIDE/EXTRA PRACTICE PART 1: PRIME OR COMPOSITE? NAME TEST DATE FRACTION STUDY GUIDE/EXTRA PRACTICE PART 1: PRIME OR COMPOSITE? A prime number is a number that has exactly 2 factors, one and itself. Examples: 2, 3, 5, 11, 31 2 is the only even number

More information

There are 8000 registered voters in Brownsville, and 3 8. of these voters live in

There are 8000 registered voters in Brownsville, and 3 8. of these voters live in Politics and the political process affect everyone in some way. In local, state or national elections, registered voters make decisions about who will represent them and make choices about various ballot

More information

The Euclidean Algorithm

The Euclidean Algorithm The Euclidean Algorithm A METHOD FOR FINDING THE GREATEST COMMON DIVISOR FOR TWO LARGE NUMBERS To be successful using this method you have got to know how to divide. If this is something that you have

More information

Adding and Subtracting Fractions. 1. The denominator of a fraction names the fraction. It tells you how many equal parts something is divided into.

Adding and Subtracting Fractions. 1. The denominator of a fraction names the fraction. It tells you how many equal parts something is divided into. Tallahassee Community College Adding and Subtracting Fractions Important Ideas:. The denominator of a fraction names the fraction. It tells you how many equal parts something is divided into.. The numerator

More information

Session 6 Number Theory

Session 6 Number Theory Key Terms in This Session Session 6 Number Theory Previously Introduced counting numbers factor factor tree prime number New in This Session composite number greatest common factor least common multiple

More information

5-4 Prime and Composite Numbers

5-4 Prime and Composite Numbers 5-4 Prime and Composite Numbers Prime and Composite Numbers Prime Factorization Number of Divisorss Determining if a Number is Prime More About Primes Prime and Composite Numbers Students should recognizee

More information

COMPASS Numerical Skills/Pre-Algebra Preparation Guide. Introduction Operations with Integers Absolute Value of Numbers 13

COMPASS Numerical Skills/Pre-Algebra Preparation Guide. Introduction Operations with Integers Absolute Value of Numbers 13 COMPASS Numerical Skills/Pre-Algebra Preparation Guide Please note that the guide is for reference only and that it does not represent an exact match with the assessment content. The Assessment Centre

More information

Lesson 4. Factors and Multiples. Objectives

Lesson 4. Factors and Multiples. Objectives Student Name: Date: Contact Person Name: Phone Number: Lesson 4 Factors and Multiples Objectives Understand what factors and multiples are Write a number as a product of its prime factors Find the greatest

More information

Improper Fractions and Mixed Numbers

Improper Fractions and Mixed Numbers This assignment includes practice problems covering a variety of mathematical concepts. Do NOT use a calculator in this assignment. The assignment will be collected on the first full day of class. All

More information

Introduction to Fractions

Introduction to Fractions Introduction to Fractions Fractions represent parts of a whole. The top part of a fraction is called the numerator, while the bottom part of a fraction is called the denominator. The denominator states

More information

Chapter 11 Number Theory

Chapter 11 Number Theory Chapter 11 Number Theory Number theory is one of the oldest branches of mathematics. For many years people who studied number theory delighted in its pure nature because there were few practical applications

More information

Grade 6 Math Circles March 10/11, 2015 Prime Time Solutions

Grade 6 Math Circles March 10/11, 2015 Prime Time Solutions Faculty of Mathematics Waterloo, Ontario N2L 3G1 Centre for Education in Mathematics and Computing Lights, Camera, Primes! Grade 6 Math Circles March 10/11, 2015 Prime Time Solutions Today, we re going

More information

Prime Numbers and Divisibility Section 2.1 Prime Number Any whole number that has exactly two factors, 1 and itself. numbers.

Prime Numbers and Divisibility Section 2.1 Prime Number Any whole number that has exactly two factors, 1 and itself. numbers. 2 Summary DEFINITION/PROCEDURE EXAMPLE REFERENCE Prime Numbers and Divisibility Section 2. Prime Number Any whole number that has exactly two factors, and itself. 7, 3, 29, and 73 are prime numbers. p.

More information

Elementary Algebra. Section 0.4 Factors

Elementary Algebra. Section 0.4 Factors Section 0.4 Contents: Definitions: Multiplication Primes and Composites Rules of Composite Prime Factorization Answers Focus Exercises THE MULTIPLICATION TABLE x 1 2 3 4 5 6 7 8 9 10 11 12 1 1 2 3 4 5

More information

Grade 7 & 8 Math Circles October 19, 2011 Prime Numbers

Grade 7 & 8 Math Circles October 19, 2011 Prime Numbers 1 University of Waterloo Faculty of Mathematics Centre for Education in Mathematics and Computing Grade 7 & 8 Math Circles October 19, 2011 Prime Numbers Factors Definition: A factor of a number is a whole

More information

Factor Trees. Objective To provide experiences with finding the greatest common factor and the least common multiple of two numbers.

Factor Trees. Objective To provide experiences with finding the greatest common factor and the least common multiple of two numbers. Factor Trees Objective To provide experiences with finding the greatest common factor and the least common multiple of two numbers. www.everydaymathonline.com epresentations etoolkit Algorithms Practice

More information

Fractions. If the top and bottom numbers of a fraction are the same then you have a whole one.

Fractions. If the top and bottom numbers of a fraction are the same then you have a whole one. What do fractions mean? Fractions Academic Skills Advice Look at the bottom of the fraction first this tells you how many pieces the shape (or number) has been cut into. Then look at the top of the fraction

More information

Course notes on Number Theory

Course notes on Number Theory Course notes on Number Theory In Number Theory, we make the decision to work entirely with whole numbers. There are many reasons for this besides just mathematical interest, not the least of which is that

More information

Section R.2. Fractions

Section R.2. Fractions Section R.2 Fractions Learning objectives Fraction properties of 0 and 1 Writing equivalent fractions Writing fractions in simplest form Multiplying and dividing fractions Adding and subtracting fractions

More information

TYPES OF NUMBERS. Example 2. Example 1. Problems. Answers

TYPES OF NUMBERS. Example 2. Example 1. Problems. Answers TYPES OF NUMBERS When two or more integers are multiplied together, each number is a factor of the product. Nonnegative integers that have exactly two factors, namely, one and itself, are called prime

More information

FRACTIONS MODULE Part I

FRACTIONS MODULE Part I FRACTIONS MODULE Part I I. Basics of Fractions II. Rewriting Fractions in the Lowest Terms III. Change an Improper Fraction into a Mixed Number IV. Change a Mixed Number into an Improper Fraction BMR.Fractions

More information

Saxon Math Home School Edition. September 2008

Saxon Math Home School Edition. September 2008 Saxon Math Home School Edition September 2008 Saxon Math Home School Edition Lesson 4: Comparing Whole Lesson 5: Naming Whole Through Hundreds, Dollars and Cent Lesson 7: Writing and Comparing Through

More information

1.4 Factors and Prime Factorization

1.4 Factors and Prime Factorization 1.4 Factors and Prime Factorization Recall from Section 1.2 that the word factor refers to a number which divides into another number. For example, 3 and 6 are factors of 18 since 3 6 = 18. Note also that

More information

1.5 Greatest Common Factor and Least Common Multiple

1.5 Greatest Common Factor and Least Common Multiple 1.5 Greatest Common Factor and Least Common Multiple This chapter will conclude with two topics which will be used when working with fractions. Recall that factors of a number are numbers that divide into

More information

Day One: Least Common Multiple

Day One: Least Common Multiple Grade Level/Course: 5 th /6 th Grade Math Lesson/Unit Plan Name: Using Prime Factors to find LCM and GCF. Rationale/Lesson Abstract: The objective of this two- part lesson is to give students a clear understanding

More information

Prime Time: Homework Examples from ACE

Prime Time: Homework Examples from ACE Prime Time: Homework Examples from ACE Investigation 1: Building on Factors and Multiples, ACE #8, 28 Investigation 2: Common Multiples and Common Factors, ACE #11, 16, 17, 28 Investigation 3: Factorizations:

More information

3.1. RATIONAL EXPRESSIONS

3.1. RATIONAL EXPRESSIONS 3.1. RATIONAL EXPRESSIONS RATIONAL NUMBERS In previous courses you have learned how to operate (do addition, subtraction, multiplication, and division) on rational numbers (fractions). Rational numbers

More information

FACTORING OUT COMMON FACTORS

FACTORING OUT COMMON FACTORS 278 (6 2) Chapter 6 Factoring 6.1 FACTORING OUT COMMON FACTORS In this section Prime Factorization of Integers Greatest Common Factor Finding the Greatest Common Factor for Monomials Factoring Out the

More information

Prime Factorization, Greatest Common Factor (GCF), and Least Common Multiple (LCM)

Prime Factorization, Greatest Common Factor (GCF), and Least Common Multiple (LCM) Prime Factorization, Greatest Common Factor (GCF), and Least Common Multiple (LCM) Definition of a Prime Number A prime number is a whole number greater than 1 AND can only be divided evenly by 1 and itself.

More information

INTRODUCTION TO FRACTIONS

INTRODUCTION TO FRACTIONS Tallahassee Community College 16 INTRODUCTION TO FRACTIONS Figure A (Use for 1 5) 1. How many parts are there in this circle?. How many parts of the circle are shaded?. What fractional part of the circle

More information

To Simplify or Not to Simplify

To Simplify or Not to Simplify Overview Activity ID: 8942 Math Concepts Materials Students will discuss fractions and what it means to simplify them Number sense TI-34 using prime factorization. Students will then use the calculator

More information

CISC - Curriculum & Instruction Steering Committee. California County Superintendents Educational Services Association

CISC - Curriculum & Instruction Steering Committee. California County Superintendents Educational Services Association CISC - Curriculum & Instruction Steering Committee California County Superintendents Educational Services Association Primary Content Module IV The Winning EQUATION NUMBER SENSE: Factors of Whole Numbers

More information

SIMPLIFYING ALGEBRAIC FRACTIONS

SIMPLIFYING ALGEBRAIC FRACTIONS Tallahassee Community College 5 SIMPLIFYING ALGEBRAIC FRACTIONS In arithmetic, you learned that a fraction is in simplest form if the Greatest Common Factor (GCF) of the numerator and the denominator is

More information

MODULAR ARITHMETIC. a smallest member. It is equivalent to the Principle of Mathematical Induction.

MODULAR ARITHMETIC. a smallest member. It is equivalent to the Principle of Mathematical Induction. MODULAR ARITHMETIC 1 Working With Integers The usual arithmetic operations of addition, subtraction and multiplication can be performed on integers, and the result is always another integer Division, on

More information

Multiplying and Dividing Fractions

Multiplying and Dividing Fractions Multiplying and Dividing Fractions 1 Overview Fractions and Mixed Numbers Factors and Prime Factorization Simplest Form of a Fraction Multiplying Fractions and Mixed Numbers Dividing Fractions and Mixed

More information

How do you compare numbers? On a number line, larger numbers are to the right and smaller numbers are to the left.

How do you compare numbers? On a number line, larger numbers are to the right and smaller numbers are to the left. The verbal answers to all of the following questions should be memorized before completion of pre-algebra. Answers that are not memorized will hinder your ability to succeed in algebra 1. Number Basics

More information

Chapter 15 Radical Expressions and Equations Notes

Chapter 15 Radical Expressions and Equations Notes Chapter 15 Radical Expressions and Equations Notes 15.1 Introduction to Radical Expressions The symbol is called the square root and is defined as follows: a = c only if c = a Sample Problem: Simplify

More information

Pre-Calculus II Factoring and Operations on Polynomials

Pre-Calculus II Factoring and Operations on Polynomials Factoring... 1 Polynomials...1 Addition of Polynomials... 1 Subtraction of Polynomials...1 Multiplication of Polynomials... Multiplying a monomial by a monomial... Multiplying a monomial by a polynomial...

More information

Math 201: Homework 7 Solutions

Math 201: Homework 7 Solutions Math 201: Homework 7 Solutions 1. ( 5.2 #4) (a) The factors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72. (b) The factors of 81 are 1, 3, 9, 27, and 81. (c) The factors of 62 are 1, 2, 31, and

More information

Numerator Denominator

Numerator Denominator Fractions A fraction is any part of a group, number or whole. Fractions are always written as Numerator Denominator A unitary fraction is one where the numerator is always 1 e.g 1 1 1 1 1...etc... 2 3

More information

Tips, tricks and formulae on H.C.F and L.C.M. Follow the steps below to find H.C.F of given numbers by prime factorization method.

Tips, tricks and formulae on H.C.F and L.C.M. Follow the steps below to find H.C.F of given numbers by prime factorization method. Highest Common Factor (H.C.F) Tips, tricks and formulae on H.C.F and L.C.M H.C.F is the highest common factor or also known as greatest common divisor, the greatest number which exactly divides all the

More information

PREPARATION FOR MATH TESTING at CityLab Academy

PREPARATION FOR MATH TESTING at CityLab Academy PREPARATION FOR MATH TESTING at CityLab Academy compiled by Gloria Vachino, M.S. Refresh your math skills with a MATH REVIEW and find out if you are ready for the math entrance test by taking a PRE-TEST

More information

2. Perform the division as if the numbers were whole numbers. You may need to add zeros to the back of the dividend to complete the division

2. Perform the division as if the numbers were whole numbers. You may need to add zeros to the back of the dividend to complete the division Math Section 5. Dividing Decimals 5. Dividing Decimals Review from Section.: Quotients, Dividends, and Divisors. In the expression,, the number is called the dividend, is called the divisor, and is called

More information

This section demonstrates some different techniques of proving some general statements.

This section demonstrates some different techniques of proving some general statements. Section 4. Number Theory 4.. Introduction This section demonstrates some different techniques of proving some general statements. Examples: Prove that the sum of any two odd numbers is even. Firstly you

More information

Lesson 3.1 Factors and Multiples of Whole Numbers Exercises (pages 140 141)

Lesson 3.1 Factors and Multiples of Whole Numbers Exercises (pages 140 141) Lesson 3.1 Factors and Multiples of Whole Numbers Exercises (pages 140 141) A 3. Multiply each number by 1, 2, 3, 4, 5, and 6. a) 6 1 = 6 6 2 = 12 6 3 = 18 6 4 = 24 6 5 = 30 6 6 = 36 So, the first 6 multiples

More information

Recall the process used for adding decimal numbers. 1. Place the numbers to be added in vertical format, aligning the decimal points.

Recall the process used for adding decimal numbers. 1. Place the numbers to be added in vertical format, aligning the decimal points. 2 MODULE 4. DECIMALS 4a Decimal Arithmetic Adding Decimals Recall the process used for adding decimal numbers. Adding Decimals. To add decimal numbers, proceed as follows: 1. Place the numbers to be added

More information

6th Grade Vocabulary Words

6th Grade Vocabulary Words 1. sum the answer when you add Ex: 3 + 9 = 12 12 is the sum 2. difference the answer when you subtract Ex: 17-9 = 8 difference 8 is the 3. the answer when you multiply Ex: 7 x 8 = 56 56 is the 4. quotient

More information

Module 2: Working with Fractions and Mixed Numbers. 2.1 Review of Fractions. 1. Understand Fractions on a Number Line

Module 2: Working with Fractions and Mixed Numbers. 2.1 Review of Fractions. 1. Understand Fractions on a Number Line Module : Working with Fractions and Mixed Numbers.1 Review of Fractions 1. Understand Fractions on a Number Line Fractions are used to represent quantities between the whole numbers on a number line. A

More information

5.1 FACTORING OUT COMMON FACTORS

5.1 FACTORING OUT COMMON FACTORS C H A P T E R 5 Factoring he sport of skydiving was born in the 1930s soon after the military began using parachutes as a means of deploying troops. T Today, skydiving is a popular sport around the world.

More information

Primes. Name Period Number Theory

Primes. Name Period Number Theory Primes Name Period A Prime Number is a whole number whose only factors are 1 and itself. To find all of the prime numbers between 1 and 100, complete the following exercise: 1. Cross out 1 by Shading in

More information

Session 29 Scientific Notation and Laws of Exponents. If you have ever taken a Chemistry class, you may have encountered the following numbers:

Session 29 Scientific Notation and Laws of Exponents. If you have ever taken a Chemistry class, you may have encountered the following numbers: Session 9 Scientific Notation and Laws of Exponents If you have ever taken a Chemistry class, you may have encountered the following numbers: There are approximately 60,4,79,00,000,000,000,000 molecules

More information

6.1 The Greatest Common Factor; Factoring by Grouping

6.1 The Greatest Common Factor; Factoring by Grouping 386 CHAPTER 6 Factoring and Applications 6.1 The Greatest Common Factor; Factoring by Grouping OBJECTIVES 1 Find the greatest common factor of a list of terms. 2 Factor out the greatest common factor.

More information

An Introduction to Number Theory Prime Numbers and Their Applications.

An Introduction to Number Theory Prime Numbers and Their Applications. East Tennessee State University Digital Commons @ East Tennessee State University Electronic Theses and Dissertations 8-2006 An Introduction to Number Theory Prime Numbers and Their Applications. Crystal

More information

Factorizations: Searching for Factor Strings

Factorizations: Searching for Factor Strings " 1 Factorizations: Searching for Factor Strings Some numbers can be written as the product of several different pairs of factors. For example, can be written as 1, 0,, 0, and. It is also possible to write

More information

Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any.

Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. Algebra 2 - Chapter Prerequisites Vocabulary Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. P1 p. 1 1. counting(natural) numbers - {1,2,3,4,...}

More information

Simplifying Algebraic Fractions

Simplifying Algebraic Fractions 5. Simplifying Algebraic Fractions 5. OBJECTIVES. Find the GCF for two monomials and simplify a fraction 2. Find the GCF for two polynomials and simplify a fraction Much of our work with algebraic fractions

More information

Reteaching. Properties of Operations

Reteaching. Properties of Operations - Properties of Operations The commutative properties state that changing the order of addends or factors in a multiplication or addition expression does not change the sum or the product. Examples: 5

More information

A.4 Polynomial Division; Synthetic Division

A.4 Polynomial Division; Synthetic Division SECTION A.4 Polynomial Division; Synthetic Division 977 A.4 Polynomial Division; Synthetic Division OBJECTIVES 1 Divide Polynomials Using Long Division 2 Divide Polynomials Using Synthetic Division 1 Divide

More information

ARITHMETIC. Overview. Testing Tips

ARITHMETIC. Overview. Testing Tips ARITHMETIC Overview The Arithmetic section of ACCUPLACER contains 17 multiple choice questions that measure your ability to complete basic arithmetic operations and to solve problems that test fundamental

More information

Simplifying Fractions

Simplifying Fractions . Simplifying Fractions. OBJECTIVES 1. Determine whether two fractions are equivalent. Use the fundamental principle to simplify fractions It is possible to represent the same portion of the whole by different

More information

Clifton High School Mathematics Summer Workbook Algebra 1

Clifton High School Mathematics Summer Workbook Algebra 1 1 Clifton High School Mathematics Summer Workbook Algebra 1 Completion of this summer work is required on the first day of the school year. Date Received: Date Completed: Student Signature: Parent Signature:

More information

MATH 4D October 4, 2015 HOMEWORK 3

MATH 4D October 4, 2015 HOMEWORK 3 MATH 4D October 4, 2015 HOMEWORK 3 1. A package of plastic forks contains 16 forks. A package of plastic knives contains 12 knives. What is the smallest number of packages of each kind you have to buy

More information

Mathematics of Cryptography

Mathematics of Cryptography CHAPTER 2 Mathematics of Cryptography Part I: Modular Arithmetic, Congruence, and Matrices Objectives This chapter is intended to prepare the reader for the next few chapters in cryptography. The chapter

More information

Section 5.5 Dividing Decimals

Section 5.5 Dividing Decimals Section 5.5 Dividing Decimals Objectives In this section, you will learn to: To successfully complete this section, you need to understand: Divide decimal numbers by whole numbers. Dividing whole numbers

More information

Lowest Common Multiple and Highest Common Factor

Lowest Common Multiple and Highest Common Factor Lowest Common Multiple and Highest Common Factor Multiple: The multiples of a number are its times table If you want to find out if a number is a multiple of another number you just need to divide the

More information

Objectives: In this fractions unit, we will

Objectives: In this fractions unit, we will Objectives: In this fractions unit, we will subtract and add fractions with unlike denominators (pp. 101 102) multiply fractions(pp.109 110) Divide fractions (pp. 113 114) Review: Which is greater, 1/3

More information

Solution Guide Chapter 14 Mixing Fractions, Decimals, and Percents Together

Solution Guide Chapter 14 Mixing Fractions, Decimals, and Percents Together Solution Guide Chapter 4 Mixing Fractions, Decimals, and Percents Together Doing the Math from p. 80 2. 0.72 9 =? 0.08 To change it to decimal, we can tip it over and divide: 9 0.72 To make 0.72 into a

More information

6.1. The Set of Fractions

6.1. The Set of Fractions CHAPTER 6 Fractions 6.1. The Set of Fractions Problem (Page 216). A child has a set of 10 cubical blocks. The lengths of the edges are 1 cm, 2 cm, cm,..., 10 cm. Using all the cubes, can the child build

More information

Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.

Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem. Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem. Solve word problems that call for addition of three whole numbers

More information

Tests for Divisibility, Theorems for Divisibility, the Prime Factor Test

Tests for Divisibility, Theorems for Divisibility, the Prime Factor Test 1 Tests for Divisibility, Theorems for Divisibility, the Prime Factor Test Definition: Prime numbers are numbers with only two factors, one and itself. For example: 2, 3, and 5. Definition: Composite numbers

More information

Math Workshop October 2010 Fractions and Repeating Decimals

Math Workshop October 2010 Fractions and Repeating Decimals Math Workshop October 2010 Fractions and Repeating Decimals This evening we will investigate the patterns that arise when converting fractions to decimals. As an example of what we will be looking at,

More information

Chapter R.4 Factoring Polynomials

Chapter R.4 Factoring Polynomials Chapter R.4 Factoring Polynomials Introduction to Factoring To factor an expression means to write the expression as a product of two or more factors. Sample Problem: Factor each expression. a. 15 b. x

More information

Now that we have a handle on the integers, we will turn our attention to other types of numbers.

Now that we have a handle on the integers, we will turn our attention to other types of numbers. 1.2 Rational Numbers Now that we have a handle on the integers, we will turn our attention to other types of numbers. We start with the following definitions. Definition: Rational Number- any number that

More information

FACTORS AND MULTIPLES Answer Key

FACTORS AND MULTIPLES Answer Key I. Find prime factors by factor tree method FACTORS AND MULTIPLES Answer Key a. 768 2 384 2 192 2 96 2 48 2 24 2 12 2 6 2 3 768 = 2*2*2*2*2*2*2*2 *3 b. 1608 3 536 2 268 2 134 2 67 1608 = 3*2*2*2*67 c.

More information

MTH 231 Practice Test SKILLS Problems (Sections 3.3, 3.4, 4.1, 4.2, 5.1, 5.2) Provide an appropriate response.

MTH 231 Practice Test SKILLS Problems (Sections 3.3, 3.4, 4.1, 4.2, 5.1, 5.2) Provide an appropriate response. MTH 231 Practice Test SKILLS Problems (Sections 3.3, 3.4, 4.1, 4.2, 5.1, 5.2) Calculate / demonstrate using the expanded algorithm. Then do the same problem using the standard algorithm. 1) 72 + 806 A)

More information

15 Prime and Composite Numbers

15 Prime and Composite Numbers 15 Prime and Composite Numbers Divides, Divisors, Factors, Multiples In section 13, we considered the division algorithm: If a and b are whole numbers with b 0 then there exist unique numbers q and r such

More information

Accuplacer Arithmetic Study Guide

Accuplacer Arithmetic Study Guide Accuplacer Arithmetic Study Guide Section One: Terms Numerator: The number on top of a fraction which tells how many parts you have. Denominator: The number on the bottom of a fraction which tells how

More information

ACCUPLACER Arithmetic Assessment Preparation Guide

ACCUPLACER Arithmetic Assessment Preparation Guide ACCUPLACER Arithmetic Assessment Preparation Guide Please note that the guide is for reference only and that it does not represent an exact match with the assessment content. The Assessment Centre at George

More information

SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics. Numbers

SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics. Numbers SCHOOL OF ENGINEERING & BUILT ENVIRONMENT Mathematics Numbers. Factors 2. Multiples 3. Prime and Composite Numbers 4. Modular Arithmetic 5. Boolean Algebra 6. Modulo 2 Matrix Arithmetic 7. Number Systems

More information

HOSPITALITY Math Assessment Preparation Guide. Introduction Operations with Whole Numbers Operations with Integers 9

HOSPITALITY Math Assessment Preparation Guide. Introduction Operations with Whole Numbers Operations with Integers 9 HOSPITALITY Math Assessment Preparation Guide Please note that the guide is for reference only and that it does not represent an exact match with the assessment content. The Assessment Centre at George

More information

Strengthening Multiplicative Reasoning with Prime Numbers

Strengthening Multiplicative Reasoning with Prime Numbers Strengthening Multiplicative Reasoning with Prime Numbers Matt Roscoe University of Montana matt.roscoe@umontana.edu Ziv Feldman Boston University zfeld@bu.edu NCTM Research Conference San Francisco, CA

More information

4 th Grade. Math Common Core I Can Checklists

4 th Grade. Math Common Core I Can Checklists 4 th Grade Math Common Core I Can Checklists Math Common Core Operations and Algebraic Thinking Use the four operations with whole numbers to solve problems. 1. I can interpret a multiplication equation

More information

Unit 1, Review Transitioning from Previous Mathematics Instructional Resources: McDougal Littell: Course 1

Unit 1, Review Transitioning from Previous Mathematics Instructional Resources: McDougal Littell: Course 1 Unit 1, Review Transitioning from Previous Mathematics Transitioning from previous mathematics to Sixth Grade Mathematics Understand the relationship between decimals, fractions and percents and demonstrate

More information

The Teacher s Circle Number Theory, Part 1 Joshua Zucker, August 14, 2006

The Teacher s Circle Number Theory, Part 1 Joshua Zucker, August 14, 2006 The Teacher s Circle Number Theory, Part 1 Joshua Zucker, August 14, 2006 joshua.zucker@stanfordalumni.org [A few words about MathCounts and its web site http://mathcounts.org at some point.] Number theory

More information

Simply Math. Everyday Math Skills NWT Literacy Council

Simply Math. Everyday Math Skills NWT Literacy Council Simply Math Everyday Math Skills 2009 NWT Literacy Council Acknowledgement The NWT Literacy Council gratefully acknowledges the financial assistance for this project from the Department of Education, Culture

More information

DECIMALS are special fractions whose denominators are powers of 10.

DECIMALS are special fractions whose denominators are powers of 10. DECIMALS DECIMALS are special fractions whose denominators are powers of 10. Since decimals are special fractions, then all the rules we have already learned for fractions should work for decimals. The

More information

Unit 1: Polynomials. Expressions: - mathematical sentences with no equal sign. Example: 3x + 2

Unit 1: Polynomials. Expressions: - mathematical sentences with no equal sign. Example: 3x + 2 Pure Math 0 Notes Unit : Polynomials Unit : Polynomials -: Reviewing Polynomials Epressions: - mathematical sentences with no equal sign. Eample: Equations: - mathematical sentences that are equated with

More information

3.4 Multiplication and Division of Rational Numbers

3.4 Multiplication and Division of Rational Numbers 3.4 Multiplication and Division of Rational Numbers We now turn our attention to multiplication and division with both fractions and decimals. Consider the multiplication problem: 8 12 2 One approach is

More information

Click on the links below to jump directly to the relevant section

Click on the links below to jump directly to the relevant section Click on the links below to jump directly to the relevant section Basic review Writing fractions in simplest form Comparing fractions Converting between Improper fractions and whole/mixed numbers Operations

More information

Prime Numbers and Divisibility

Prime Numbers and Divisibility 2.1 Prime Numbers and Divisibility 2.1 OBJECTIVES 1. Find the factors of a number 2. Determine whether a number is prime, composite, or neither 3. Determine whether a number is divisible by 2, 3, or 5

More information

SIXTH GRADE MATH. Quarter 1

SIXTH GRADE MATH. Quarter 1 Quarter 1 SIXTH GRADE MATH Numeration - Place value - Comparing and ordering whole numbers - Place value of exponents - place value of decimals - multiplication and division by 10, 100, 1,000 - comparing

More information