MATH 112 Section 4.3: GCDs and LCMs
|
|
- Barrie Bennett
- 7 years ago
- Views:
Transcription
1 MATH 112 Section 4.3: GCDs and LCMs Prof. Jonathan Duncan Walla Walla College Fall Quarter, 2006
2 Outline 1 Greatest Common Factors 2 Least Common Multiples 3 Relationships Between GCFs and LCMs 4 Conclusion
3 A Motivating In our last section in chapter 4, we will examine two different numbers which can be built from the factors of a pair of given numbers. A quilter is using a rectangular piece of cloth 300 inches by 90 inches in size. He wishes to make a large quilt out of perfect squares. What are the dimensions of the largest possible square (in whole inches) which will exactly use up the fabric? We need an x by x square where x is a divisor of both 300 and 90. How do we go about finding x?
4 The Greatest Common Factor Greatest Common Factor The greatest common factor of two numbers a and b, written as GCF (a, b), is the largest number which is a factor of both a and b. Properties of the GCF The GCF of a and b has the following properties: It is a factor of both a and b, so it is less than both. Prime numbers a and b have a GCF of 1. Even numbers a and b have an even GCF of 2 or more. Odd numbers a and b have an odd GCF. Modeling the GCF The greatest common factor can be modeled using cuisinare rods. Use rods to model GCF (8, 12).
5 Finding the GCF by Listing Factors There are several ways we can try to find Greatest Common Factors. One of the most basic methods is to list factors. Listing Factors To find GCF (a, b) list all factors of a and all factors of b and find the largest number common to both lists. Find GCF (84, 126) by listing factors. Advantages and Disadvantages Advantages: Visual and convincing Disadvantages: time consuming, hard for larger numbers, lots of extra information required.
6 Finding the GCF by Intuition Sometimes it is possible to just know the answer to a question based on previous experience. Intuition To find GCF (a, b) think about the factors of a and b and see if the largest common factor comes to you. Use intuition to find GCF (64, 12). Advantages and Disadvantages Advantages: it is fast! Disadvantages: it is hard for large numbers, of questionable accuracy, and does not work for everybody
7 Finding the GCF by Repeated Division If you don t have good intuition, there is still a better method to find the GCF than listing out all the factors. Repeated Division To find GCF (a, b) using repeated division, repeatedly divide a and b by common prime factors until this can no longer be done. Then, the GCF is the product of the common prime factors. Use repeated division to find GCF (135, 75). Advantages and Disadvantages Advantages: systematic and relatively quick Disadvantages: involves division and recognizing common prime factors.
8 Finding the GCF by Prime Factorization The last method we will examine for finding the GCF is by using prime factorizations. It is in many ways similar to the repeated division and listing of factors methods seen earlier. Prime Factorization To find GCF (a, b), write the prime factorization of a and b. The GCF is the product of all prime factors common to a and b. Use prime factorizations to find GCF (84, 126) Advantages and Disadvantages Advantages: systematic and relatively quick Disadvantages: requires complete factorization
9 Another Important Number Another important number which appears in many computations has to do with common multiples instead of common factors. Your school is having a fund raiser at which your class is responsible for selling vegie-burgers. When you go to the store to buy supplies, you find that the patties come un packs of 12, but the buns come in packs of 8. What is the smallest number of each you could purchase so that you have the same number of patties as buns? We need a number m which is something times 8 and something else times 12 and as small as possible.
10 The Least Common Multiple The Least Common Multiple The least common multiple of two numbers a and b, written LCM(a, b), is the smallest number which is both a multiple of a and a multiple of b. Properties of the LCM The LCM of a and b has the following properties: It is a multiple of both a and b, so at least as big as both. It is less than or equal to a b. Even numbers a and b have an even GCF. Odd numbers a and b have an odd GCF. Modeling the LCM The least common multiple can be modeled using cuisinare rods. Use rods to model LCM(4, 6).
11 Finding the LCM by Listing Multiples As with the greatest common factor, the simplest way to approach this problem may be to just start listing multiples. Listing Multiples To find LCM(a, b) first list multiples of a up to a b and multiples of b up to a b. The smallest multiple common to both lists is the LCM. Find LCM(12, 30). Advantages and Disadvantages What are some of the advantages and disadvantages of this method of finding the LCM?
12 Finding the LCM by Prime Factorization As with the GCF, perhaps the best method for finding the LCM is through the use of prime factorization. Prime Factorization To find LCM(a, b) first write the prime factorization of a and b. Then look for the shortest list of primes which contains all factors in both the list for a and b including repeated primes. The product of this list is the LCM. Find LCM(12, 30). Advantages and Disadvantages What are some of the advantages and disadvantages of this method of finding the LCM?
13 The GCF and LCM The prime factorization method for finding the GCF and LCM of two numbers is somewhat similar. Let s examine some of the relationships between GCFs and LCMs, starting with an example. Find both the GCF and LCM of 504 and 98 There are several things we can do to solve this problem: Use prime factorizations Use a Venn Diagram of factors
14 Relating the GCF and LCM There is a relationship between the GCF and LCM of two numbers due to their prime factorizations. Find the GCF and LCM of a and b. The General Relationship a b GCF (a, b) LCM(a, b) In general, a b = GCF (a, b) LCM(a, b), or stated another way, LCM(a, b) = a b GCF (a,b).
15 Some Closing s Sometimes using the relationships between GCFs and LCMs can help in finding either or both of these numbers. Find LCM(1485, 825). If GCF (45, x) = 9 and LCM(45, x) = 135 find x.
16 Important Concepts Things to Remember from Section The definition of a greatest common factor 2 How to find greatest common factors 3 The definition of a least common multiple 4 How to find least common multiples 5 Relationships between GCFs and LCMs
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 informationGrade 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 information17 Greatest Common Factors and Least Common Multiples
17 Greatest Common Factors and Least Common Multiples Consider the following concrete problem: An architect is designing an elegant display room for art museum. One wall is to be covered with large square
More informationGrade 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 informationGreatest 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 informationACTIVITY: Identifying Common Multiples
1.6 Least Common Multiple of two numbers? How can you find the least common multiple 1 ACTIVITY: Identifying Common Work with a partner. Using the first several multiples of each number, copy and complete
More informationFactoring Whole Numbers
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
More informationPrime 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 informationSession 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 informationFactors and Products
CHAPTER 3 Factors and Products What You ll Learn use different strategies to find factors and multiples of whole numbers identify prime factors and write the prime factorization of a number find square
More informationChapter 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 informationFACTORS, PRIME NUMBERS, H.C.F. AND L.C.M.
Mathematics Revision Guides Factors, Prime Numbers, H.C.F. and L.C.M. Page 1 of 16 M.K. HOME TUITION Mathematics Revision Guides Level: GCSE Higher Tier FACTORS, PRIME NUMBERS, H.C.F. AND L.C.M. Version:
More informationCISC - 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 informationLesson 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 information1. When the least common multiple of 8 and 20 is multiplied by the greatest common factor of 8 and 20, what is the result?
Black Equivalent Fractions and LCM 1. When the least common multiple of 8 and 20 is multiplied by the greatest common factor of 8 and 20, what is the result? 2. The sum of three consecutive integers is
More informationPreviously, 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 informationDay 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 informationCommon Multiples. List the multiples of 3. The multiples of 3 are 3 1, 3 2, 3 3, 3 4,...
.2 Common Multiples.2 OBJECTIVES 1. Find the least common multiple (LCM) of two numbers 2. Find the least common multiple (LCM) of a group of numbers. Compare the size of two fractions In this chapter,
More informationCalculate Highest Common Factors(HCFs) & Least Common Multiples(LCMs) NA1
Calculate Highest Common Factors(HCFs) & Least Common Multiples(LCMs) NA1 What are the multiples of 5? The multiples are in the five times table What are the factors of 90? Each of these is a pair of factors.
More informationPrime 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 information1.5 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 informationFactorizations: 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 informationToday s Topics. Primes & Greatest Common Divisors
Today s Topics Primes & Greatest Common Divisors Prime representations Important theorems about primality Greatest Common Divisors Least Common Multiples Euclid s algorithm Once and for all, what are prime
More informationPrimes. 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 informationThe 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 informationThe GMAT Guru. Prime Factorization: Theory and Practice
. Prime Factorization: Theory and Practice The following is an ecerpt from The GMAT Guru Guide, available eclusively to clients of The GMAT Guru. If you would like more information about GMAT Guru services,
More informationAdding 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 informationPrime 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 informationLesson 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 informationAND LEAST COMMON MULTIPLE
106 Chapter 4 Number Theory Activity Set 4.2 MODELS FOR GREATEST COMMON FACTOR AND LEAST COMMON MULTIPLE PURPOSE To use a linear model to illustrate the concepts of greatest common factor and least common
More informationGreatest Common Factors and Least Common Multiples with Venn Diagrams
Greatest Common Factors and Least Common Multiples with Venn Diagrams Stephanie Kolitsch and Louis Kolitsch The University of Tennessee at Martin Martin, TN 38238 Abstract: In this article the authors
More informationLowest 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 information6.1 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 informationGreatest Common Factor
SKILL 10 Name Greatest Common Factor Date The greatest common factor (GCF) of two or more numbers is the greatest number that is a factor of each number. One way to find the greatest common factor is to
More informationCHAPTER 5. Number Theory. 1. Integers and Division. Discussion
CHAPTER 5 Number Theory 1. Integers and Division 1.1. Divisibility. Definition 1.1.1. Given two integers a and b we say a divides b if there is an integer c such that b = ac. If a divides b, we write a
More informationChapter 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 informationMATH 22. THE FUNDAMENTAL THEOREM of ARITHMETIC. Lecture R: 10/30/2003
MATH 22 Lecture R: 10/30/2003 THE FUNDAMENTAL THEOREM of ARITHMETIC You must remember this, A kiss is still a kiss, A sigh is just a sigh; The fundamental things apply, As time goes by. Herman Hupfeld
More informationNF5-12 Flexibility with Equivalent Fractions and Pages 110 112
NF5- Flexibility with Equivalent Fractions and Pages 0 Lowest Terms STANDARDS preparation for 5.NF.A., 5.NF.A. Goals Students will equivalent fractions using division and reduce fractions to lowest terms.
More informationAn 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 informationMATHCOUNTS TOOLBOX Facts, Formulas and Tricks
MATHCOUNTS TOOLBOX Facts, Formulas and Tricks MATHCOUNTS Coaching Kit 40 I. PRIME NUMBERS from 1 through 100 (1 is not prime!) 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 II.
More informationAlgebra 1: Basic Skills Packet Page 1 Name: Integers 1. 54 + 35 2. 18 ( 30) 3. 15 ( 4) 4. 623 432 5. 8 23 6. 882 14
Algebra 1: Basic Skills Packet Page 1 Name: Number Sense: Add, Subtract, Multiply or Divide without a Calculator Integers 1. 54 + 35 2. 18 ( 30) 3. 15 ( 4) 4. 623 432 5. 8 23 6. 882 14 Decimals 7. 43.21
More information. 0 1 10 2 100 11 1000 3 20 1 2 3 4 5 6 7 8 9
Introduction The purpose of this note is to find and study a method for determining and counting all the positive integer divisors of a positive integer Let N be a given positive integer We say d is a
More informationFractions. 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 informationFACTORING 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 informationFactoring Polynomials
UNIT 11 Factoring Polynomials You can use polynomials to describe framing for art. 396 Unit 11 factoring polynomials A polynomial is an expression that has variables that represent numbers. A number can
More informationSection 6.1 Factoring Expressions
Section 6.1 Factoring Expressions The first method we will discuss, in solving polynomial equations, is the method of FACTORING. Before we jump into this process, you need to have some concept of what
More informationFACTORING POLYNOMIALS
296 (5-40) Chapter 5 Exponents and Polynomials where a 2 is the area of the square base, b 2 is the area of the square top, and H is the distance from the base to the top. Find the volume of a truncated
More informationCONTENTS. Please note:
CONTENTS Introduction...iv. Number Systems... 2. Algebraic Expressions.... Factorising...24 4. Solving Linear Equations...8. Solving Quadratic Equations...0 6. Simultaneous Equations.... Long Division
More informationFACTORS 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 informationFactor 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 informationPOLYNOMIAL FUNCTIONS
POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a
More informationMATH 13150: Freshman Seminar Unit 10
MATH 13150: Freshman Seminar Unit 10 1. Relatively prime numbers and Euler s function In this chapter, we are going to discuss when two numbers are relatively prime, and learn how to count the numbers
More information3.3 Addition and Subtraction of Rational Numbers
3.3 Addition and Subtraction of Rational Numbers In this section we consider addition and subtraction of both fractions and decimals. We start with addition and subtraction of fractions with the same denominator.
More informationIntroduction to Fractions
Section 0.6 Contents: Vocabulary of Fractions A Fraction as division Undefined Values First Rules of Fractions Equivalent Fractions Building Up Fractions VOCABULARY OF FRACTIONS Simplifying Fractions Multiplying
More informationFactoring Polynomials
Factoring Polynomials Factoring Factoring is the process of writing a polynomial as the product of two or more polynomials. The factors of 6x 2 x 2 are 2x + 1 and 3x 2. In this section, we will be factoring
More information3 cups ¾ ½ ¼ 2 cups ¾ ½ ¼. 1 cup ¾ ½ ¼. 1 cup. 1 cup ¾ ½ ¼ ¾ ½ ¼. 1 cup. 1 cup ¾ ½ ¼ ¾ ½ ¼
cups cups cup Fractions are a form of division. When I ask what is / I am asking How big will each part be if I break into equal parts? The answer is. This a fraction. A fraction is part of a whole. The
More informationMath 10C. Course: Polynomial Products and Factors. Unit of Study: Step 1: Identify the Outcomes to Address. Guiding Questions:
Course: Unit of Study: Math 10C Polynomial Products and Factors Step 1: Identify the Outcomes to Address Guiding Questions: What do I want my students to learn? What can they currently understand and do?
More informationPrime Numbers A prime number is a whole number, greater than 1, that has only 1 an itself as factors.
Prime Numbers A prime number is a whole number, greater than 1, that has only 1 an itself as factors. Composite Numbers A composite number is a whole number, greater than 1, that are not prime. Prime Factorization
More informationWorking with whole numbers
1 CHAPTER 1 Working with whole numbers In this chapter you will revise earlier work on: addition and subtraction without a calculator multiplication and division without a calculator using positive and
More informationMaths Workshop for Parents 2. Fractions and Algebra
Maths Workshop for Parents 2 Fractions and Algebra What is a fraction? A fraction is a part of a whole. There are two numbers to every fraction: 2 7 Numerator Denominator 2 7 This is a proper (or common)
More informationUnit 1 Number Sense. In this unit, students will study repeating decimals, percents, fractions, decimals, and proportions.
Unit 1 Number Sense In this unit, students will study repeating decimals, percents, fractions, decimals, and proportions. BLM Three Types of Percent Problems (p L-34) is a summary BLM for the material
More informationMATH10040 Chapter 2: Prime and relatively prime numbers
MATH10040 Chapter 2: Prime and relatively prime numbers Recall the basic definition: 1. Prime numbers Definition 1.1. Recall that a positive integer is said to be prime if it has precisely two positive
More informationAnchorage School District/Alaska Sr. High Math Performance Standards Algebra
Anchorage School District/Alaska Sr. High Math Performance Standards Algebra Algebra 1 2008 STANDARDS PERFORMANCE STANDARDS A1:1 Number Sense.1 Classify numbers as Real, Irrational, Rational, Integer,
More informationThe last three chapters introduced three major proof techniques: direct,
CHAPTER 7 Proving Non-Conditional Statements The last three chapters introduced three major proof techniques: direct, contrapositive and contradiction. These three techniques are used to prove statements
More informationVolume of Pyramids and Cones
Volume of Pyramids and Cones Objective To provide experiences with investigating the relationships between the volumes of geometric solids. www.everydaymathonline.com epresentations etoolkit Algorithms
More informationBlack Problems - Prime Factorization, Greatest Common Factor and Simplifying Fractions
Black Problems Prime Factorization, Greatest Common Factor and Simplifying Fractions A natural number n, such that n >, can t be written as the sum of two more consecutive odd numbers if and only if n
More informationNot for resale. 4.1 Divisibility of Natural Numbers 4.2 Tests for Divisibility 4.3 Greatest Common Divisors and Least Common Multiples
CHAPTER 4 Number Theory 4.1 Divisibility of Natural Numbers 4.2 Tests for Divisibility 4.3 Greatest Common Divisors and Least Common Multiples 4.4 Codes and Credit Card Numbers: Connections to Number Theory
More informationMATH COMPUTATION. Part 1. TIME : 15 Minutes
MATH COMPUTATION Part 1 TIME : 15 Minutes This is a practice test - the results are not valid for certificate requirements. A calculator may not be used for this test. MATH COMPUTATION 1. 182 7 = A. 20
More informationFraction Competency Packet
Fraction Competency Packet Developed by: Nancy Tufo Revised 00: Sharyn Sweeney Student Support Center North Shore Community College To use this booklet, review the glossary, study the examples, then work
More informationSPECIAL PRODUCTS AND FACTORS
CHAPTER 442 11 CHAPTER TABLE OF CONTENTS 11-1 Factors and Factoring 11-2 Common Monomial Factors 11-3 The Square of a Monomial 11-4 Multiplying the Sum and the Difference of Two Terms 11-5 Factoring the
More information6.4 Factoring Polynomials
Name Class Date 6.4 Factoring Polynomials Essential Question: What are some ways to factor a polynomial, and how is factoring useful? Resource Locker Explore Analyzing a Visual Model for Polynomial Factorization
More informationMATH 10034 Fundamental Mathematics IV
MATH 0034 Fundamental Mathematics IV http://www.math.kent.edu/ebooks/0034/funmath4.pdf Department of Mathematical Sciences Kent State University January 2, 2009 ii Contents To the Instructor v Polynomials.
More information11+ For You Maths Paper Sample Questions
This maths paper contains 70 questions and is designed to give practice on various topics. If you take 1 minute per question on average, then you are plenty fast enough! It is longer than the 40 minute
More informationSQUARE-SQUARE ROOT AND CUBE-CUBE ROOT
UNIT 3 SQUAREQUARE AND CUBEUBE (A) Main Concepts and Results A natural number is called a perfect square if it is the square of some natural number. i.e., if m = n 2, then m is a perfect square where m
More informationDigitalCommons@University of Nebraska - Lincoln
University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln MAT Exam Expository Papers Math in the Middle Institute Partnership 7-1-007 Pythagorean Triples Diane Swartzlander University
More informationGreatest Common Factor (GCF) Factoring
Section 4 4: Greatest Common Factor (GCF) Factoring The last chapter introduced the distributive process. The distributive process takes a product of a monomial and a polynomial and changes the multiplication
More informationUnit 8 Angles, 2D and 3D shapes, perimeter and area
Unit 8 Angles, 2D and 3D shapes, perimeter and area Five daily lessons Year 6 Spring term Recognise and estimate angles. Use a protractor to measure and draw acute and obtuse angles to Page 111 the nearest
More informationMath 319 Problem Set #3 Solution 21 February 2002
Math 319 Problem Set #3 Solution 21 February 2002 1. ( 2.1, problem 15) Find integers a 1, a 2, a 3, a 4, a 5 such that every integer x satisfies at least one of the congruences x a 1 (mod 2), x a 2 (mod
More informationJust the Factors, Ma am
1 Introduction Just the Factors, Ma am The purpose of this note is to find and study a method for determining and counting all the positive integer divisors of a positive integer Let N be a given positive
More informationMath 10 - Unit 3 Final Review - Numbers
Class: Date: Math 10 - Unit Final Review - Numbers Multiple Choice Identify the choice that best answers the question. 1. Write the prime factorization of 60. a. 2 7 9 b. 2 6 c. 2 2 7 d. 2 7 2. Write the
More informationMATH 90 CHAPTER 6 Name:.
MATH 90 CHAPTER 6 Name:. 6.1 GCF and Factoring by Groups Need To Know Definitions How to factor by GCF How to factor by groups The Greatest Common Factor Factoring means to write a number as product. a
More informationMATHEMATICS. Y5 Multiplication and Division 5330 Square numbers, prime numbers, factors and multiples. Equipment. MathSphere
MATHEMATICS Y5 Multiplication and Division 5330 Square numbers, prime numbers, factors and multiples Paper, pencil, ruler. Equipment MathSphere 5330 Square numbers, prime numbers, factors and multiples
More information8 Primes and Modular Arithmetic
8 Primes and Modular Arithmetic 8.1 Primes and Factors Over two millennia ago already, people all over the world were considering the properties of numbers. One of the simplest concepts is prime numbers.
More informationThe Fundamental Theorem of Arithmetic
The Fundamental Theorem of Arithmetic 1 Introduction: Why this theorem? Why this proof? One of the purposes of this course 1 is to train you in the methods mathematicians use to prove mathematical statements,
More informationThis is Factoring and Solving by Factoring, chapter 6 from the book Beginning Algebra (index.html) (v. 1.0).
This is Factoring and Solving by Factoring, chapter 6 from the book Beginning Algebra (index.html) (v. 1.0). This book is licensed under a Creative Commons by-nc-sa 3.0 (http://creativecommons.org/licenses/by-nc-sa/
More informationAll the examples in this worksheet and all the answers to questions are available as answer sheets or videos.
BIRKBECK MATHS SUPPORT www.mathsupport.wordpress.com Numbers 3 In this section we will look at - improper fractions and mixed fractions - multiplying and dividing fractions - what decimals mean and exponents
More information6.3 FACTORING ax 2 bx c WITH a 1
290 (6 14) Chapter 6 Factoring e) What is the approximate maximum revenue? f) Use the accompanying graph to estimate the price at which the revenue is zero. y Revenue (thousands of dollars) 300 200 100
More informationSIMPLIFYING 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 information15.1 Factoring Polynomials
LESSON 15.1 Factoring Polynomials Use the structure of an expression to identify ways to rewrite it. Also A.SSE.3? ESSENTIAL QUESTION How can you use the greatest common factor to factor polynomials? EXPLORE
More informationFacebook. GMAT Club CAT Tests. GMAT Toolkit ipad App. gmatclub.com/iphone. gmatclub.com/tests. The Verbal Initiative. gmatclub.
For the latest version of the GMAT ok, please visit: http://gmatclub.com/ GMAT Club s Other Resources: GMAT Club CAT Tests gmatclub.com/tests GMAT Toolkit ipad App gmatclub.com/iphone The Verbal Initiative
More informationThe Crescent Primary School Calculation Policy
The Crescent Primary School Calculation Policy Examples of calculation methods for each year group and the progression between each method. January 2015 Our Calculation Policy This calculation policy has
More information5-1 NUMBER THEORY: DIVISIBILITY; PRIME & COMPOSITE NUMBERS 210 f8
5-1 NUMBER THEORY: DIVISIBILITY; PRIME & COMPOSITE NUMBERS 210 f8 Note: Integers are the w hole numbers and their negatives (additive inverses). While our text discusses only whole numbers, all these ideas
More informationMath 132. Population Growth: the World
Math 132 Population Growth: the World S. R. Lubkin Application If you think growth in Raleigh is a problem, think a little bigger. The population of the world has been growing spectacularly fast in the
More informationFilling and Wrapping: Homework Examples from ACE
Filling and Wrapping: Homework Examples from ACE Investigation 1: Building Smart Boxes: Rectangular Prisms, ACE #3 Investigation 2: Polygonal Prisms, ACE #12 Investigation 3: Area and Circumference of
More informationFactoring Polynomials
Factoring a Polynomial Expression Factoring a polynomial is expressing the polynomial as a product of two or more factors. Simply stated, it is somewhat the reverse process of multiplying. To factor polynomials,
More informationSUNY ECC. ACCUPLACER Preparation Workshop. Algebra Skills
SUNY ECC ACCUPLACER Preparation Workshop Algebra Skills Gail A. Butler Ph.D. Evaluating Algebraic Epressions Substitute the value (#) in place of the letter (variable). Follow order of operations!!! E)
More informationFourth Grade Math Standards and "I Can Statements"
Fourth Grade Math Standards and "I Can Statements" Standard - CC.4.OA.1 Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 x 7 as a statement that 35 is 5 times as many as 7 and
More informationMultiplying 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 informationAdvanced GMAT Math Questions
Advanced GMAT Math Questions Version Quantitative Fractions and Ratios 1. The current ratio of boys to girls at a certain school is to 5. If 1 additional boys were added to the school, the new ratio of
More informationThe Prime Numbers. Definition. A prime number is a positive integer with exactly two positive divisors.
The Prime Numbers Before starting our study of primes, we record the following important lemma. Recall that integers a, b are said to be relatively prime if gcd(a, b) = 1. Lemma (Euclid s Lemma). If gcd(a,
More information