Math 201: Homework 7 Solutions


 Joanna Tyler
 2 years ago
 Views:
Transcription
1 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 62. (d) The factors of 54 are 1, 2, 3, 6, 9, 18, 27, and 54. (e) The factors of 326 are 1, 2, 163, and 326. (f) The factors of 242 are 1, 2, 11, 22, 121, and ( 5.2 #10) We begin by drawing out the list of numbers as given. Since the square root of 150 is about 12, we know we only need to cross out multiples of primes through 12. So we cross out all multiples of the primes 2, 3, 5, 7, and 11, and we are left with all the prime numbers through 150: We found the new primes 127, 131, 137, 139, and ( 5.2 #12) (a) The prime factorization of 16 is = 2 4. (b) The prime factorization of 27 is = 3 3. (c) The prime factorization of 52 is = (d) The prime factorization of 75 is = (e) The prime factorization of 112 is =
2 (f) The prime factorization of 125 is = ( 5.2 #14) We can find the top number using only the given numbers. Since the Fundamental Theorem of Arithmetic guarantees that every number has a unique prime factorization, and the factor tree we are given tells us the prime factorization of the top number, then we need only multiply the numbers up the tree to find the omitted numbers ( 5.2 #16) (a) The square root of 16 is 4, since 4 2 = 16. (b) The square root of 81 is 9, since 9 2 = 81. (c) The square root of 100 is 10, since 10 2 = 100. (d) The square root of 144 is 12, since 12 2 = ( 5.2 #18) (a) 417 is composite since, for instance, 3 divides it (note that 3 ( )). (b) 729 is composite since, for instance, 3 divides it (note that 3 ( )). (c) 1571 is prime. Note that we need only check primes through 40, since (d) 4587 is composite since, for instance 3 divides it (note that 3 ( )). (e) 35, 721 is composite since, for instance, 3 divides it (note that 3 ( )). (f) 87, 451 is composite since, for instance, 7 divides it (you can do our little splitting trick a few times, or you can just solve 7 ) 87, 451). 7. ( 5.2 #22) (a) The prime factorization of 432 is = (b) The prime factorization of 1568 is = (c) The prime factorization of 2079 is = (d) The prime factorization of 6318 is = (e) The prime factorization of 6048 is = (f) The prime factorization of 8281 is =
3 8. ( 5.2 #24) (a) False. The number 2 is even, but it is also prime. (b) False. 6 is not prime. The prime factorization of 60 would be (c) False. Both 2 and 5 are prime, and 2 5 = 10 is even. (d) True. Since the square root of 1393 is about 37, then we need only check for prime factors less than or equal to ( 5.2 #26) This question is equivalent to asking for all the factors of 24, which are 1, 2, 3, 4, 6, 8, 12, and 24. That is, you could divide the class up into 24 groups of 1 person each, 12 groups of 2 people each, 8 groups of 3 people each, 6 groups of 4 people each, 4 groups of 6 people each, 3 groups of 8 people each, 2 groups of 12 people each, or 1 big group of 24 people. 10. ( 5.2 #40) (a) We first write each given base in its prime factorization, then combine using properties of exponents = (2 2 3) 9 ( ) 15 (13 2 ) 5 = (2 2 ) (2 2 ) 15 (3 2 ) 15 (13 2 ) 5 Power of a Product Property = Power of a Power Property = Product of Powers Property (b) We first write each given base in its prime factorization, then combine using properties of exponents. 11. ( 5.3 #4) = (2 4 ) 13 (2 7) 7 (3 4 ) 14 = (2 4 ) (3 4 ) 14 Power of a Product Property = Power of a Power Property = Product of Powers Property (a) The prime factorizations of 9 and of 15 are 9 = 3 2 and 15 = 3 5 The only common factor is a single 3. Hence, GCF(9, 15) = 3. (b) The prime factorizations of 13 and of 20 are 13 = 13 and 20 = These have no common prime factor. Hence, GCF(13, 20) = 1. 3
4 (c) The prime factorizations of 6 and of 8 are 6 = 2 3 and 8 = 2 3 The only common factor is a single 2. Hence, GCF(6, 8) = 2. (d) The prime factorizations of 21 and of 63 are 21 = 3 7 and 63 = The common factors are a 3 and a 7. Hence, GCF(21, 63) = 3 7 = 21. (e) The prime factorizations of 36 and of 54 are 36 = and 54 = The common factors are a single 2 and two 3s. Hence, GCF(36, 54) = = 18. (f) The prime factorizations of 100 and of 360 are 12. ( 5.3 #6) (a) (b) 100 = and 360 = The common factors are two 2s and a single 5. Hence, GCF(100, 360) = = 20. GCF(6, 26) = GCF(6, 26 6) = GCF(6, 20) = GCF(6, 20 6) = GCF(6, 14) = GCF(6, 14 6) = GCF(6, 8) = GCF(6, 8 6) = GCF(6, 2) = 2 GCF(8, 28) = GCF(8, 28 8) = GCF(8, 20) = GCF(8, 20 8) = GCF(8, 12) = GCF(8, 12 8) = GCF(8, 4) = 4 4
5 (c) GCF(40, 56) = GCF(40, 56 40) = GCF(40, 16) = GCF(40 16, 16) = GCF(24, 16) = GCF(24 16, 16) = GCF(8, 16) = 8 (d) GCF(35, 42) = GCF(35, 42 35) = GCF(35, 7) = 7 (e) GCF(34, 85) = GCF(34, 85 34) = GCF(34, 51) = GCF(34, 51 34) = GCF(34, 17) = 17 (f) GCF(32, 55) = GCF(32, 55 32) = GCF(32, 23) = GCF(32 23, 23) = GCF(9, 23) = GCF(9, 23 9) = GCF(9, 14) = GCF(9, 14 9) = GCF(9, 5) = GCF(9 5, 5) = GCF(4, 5) = GCF(4, 5 4) = GCF(4, 1) = 1 5
6 13. ( 5.3 #8) (a) 2R48 84 ) 216 1R36 48 ) 84 1R12 36 ) 48 3R0 12 ) 36 So GCF(84, 216) = 12. (b) 1R ) 192 1R48 72 ) 120 1R24 48 ) 72 2R0 24 ) 48 (c) (d) (e) (f) (g) (h) So GCF(120, 192) = 24. So GCF(63, 84) = 21. 1R42 98 ) 140 So GCF(98, 140) = 14. So GCF(45, 90) = 45. So GCF(160, 800) = 160. So GCF(56, 588) = 28. 1R ) 935 So GCF(544, 935) = 17. 1R21 63 ) 84 10R28 56 ) 588 1R68 85 ) R14 42 ) 98 2R0 45 ) 90 5R0 160 ) 800 1R ) 544 3R0 21 ) 63 2R0 28 ) 56 1R17 68 ) 85 3R0 14 ) 42 2R ) 391 4R0 17 ) 68
7 14. ( 5.3 #10) (a) The prime factorizations of 3 and of 8 are 3 = 3 and 8 = 2 3 There are no common factors, so the diagram looks like Factors of 3 Factors of Hence, LCM(3, 8) = 3 (2 2 2) = 24. (b) The prime factorizations of 9 and of 12 are 9 = 3 2 and 12 = These only share a factor of 3, so the diagram looks like Factors of 9 Factors of Hence, LCM(9, 12) = 3 3 (2 2) = 36. (c) The prime factorizations of 12 and of 36 are 12 = and 36 = These share two 2s and a 3, so the diagram looks like Factors of 12 Factors of Hence, LCM(12, 36) = (2 2 3) 3 = 36. (d) The prime factorizations of 2 and of 9 are 2 = 2 and 9 = 3 2 7
8 There are no common factors, so the diagram looks like Factors of 2 Factors of Hence, LCM(2, 9) = 2 (3 3) = 18. (e) The prime factorizations of 24 and of 45 are 24 = and 45 = These only share the factor 3, so the diagram looks like Factors of 24 Factors of Hence, LCM(24, 45) = (2 2 2) 3 (3 5) = 360. (f) The prime factorizations of 15 and of 25 are 15 = 3 5 and 25 = 5 2 These share only a factor of 5, so the diagram looks like Factors of 15 Factors of Hence, LCM(15, 25) = = ( 5.3 #12) (a) So LCM(3, 6) = 6. 8
9 (b) (c) So LCM(4, 5) = So LCM(2, 7) = ( 5.3 #14) (a) The prime factorizations of 15 and of 20 are 15 = 3 5 and 20 = These share only a factor of 5, so the diagram looks like Factors of 15 Factors of Hence, GCF(15, 20) = 5, and LCM(15, 20) = 3 5 (2 2) = 60. (b) The prime factorizations of 50 and of 100 are 50 = and 100 = These share one factor of 2 and two factors of 5, so the diagram looks like Factors of 50 Factors of Hence, GCF(50, 100) = = 50, and LCM(50, 100) = (2 5 5) 2 =
10 (c) The prime factorizations of 24 and of 30 are 24 = and 30 = These share a factor of 2 and a factor of 3, so the diagram looks like Factors of 24 Factors of Hence, GCF(24, 30) = 2 3 = 6, and LCM(24, 30) = (2 2) (2 3) 5 = ( 5.3 #20) If the gift bags are to be identical, then you must use the same number of candles and gift cards in each bag. This is equivalent to asking for GCF(18, 24) = 6. Hence, the largest number of gift bags you can make is 6, where each contains 3 candles and 4 gift cards. 18. ( 5.3 #22) (a) The prime factorizations of 12, 18, and 24 are 12 = and 18 = and 24 = The only factors that are common to all three numbers are a single 2 and a single 3. Hence, the greatest common factor of 12, 18, and 24 is 2 3 = 6. (b) The prime factorizations of 24, 36, and 60 are 24 = and 36 = and 60 = The only factors that are common to all three numbers are two 2s and a single 3. Hence, the greatest common factor of 24, 36, and 60 is = 12. (c) The prime factorizations of 26, 52, and 78 are 19. ( 5.3 #24) 26 = 2 13 and 52 = and 78 = The only factors that are common to all three numbers are a single 2 and a single 13. Hence, the greatest common factor of 26, 52, and 78 is 2 13 = 26. (a) We ll compute the least common multiple of 4, 6, and 12 by looking at their multiples and identifying the first common one. The multiples of 4 are 4, 8, 12, 16, 20, 24,.... The multiples of 6 are 6, 12, 18, 24, 30, 36,.... The multiples of 12 are 12, 24, 36, 48, 60, 72,.... The first common multiple among these is 12, so this is our answer. 10
11 (b) We ll compute the least common multiple of 5, 16, and 20 by looking at their multiples and identifying the first common one. The multiples of 5 are 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, The multiples of 16 are 16, 32, 48, 64, 80, 96,.... The multiples of 20 are 20, 40, 60, 80, 100, 120,.... The first common multiple among these is 80, so this is our answer. (c) We ll compute the least common multiple of 24, 36, and 48 by looking at their multiples and identifying the first common one. The multiples of 24 are 24, 48, 72, 96, 120, 144,.... The multiples of 36 are 36, 72, 108, 144, 180,.... The multiples of 48 are 48, 96, 144, 192, 240,.... The first common multiple among these is 144, so this is our answer. 20. ( 5.3 #32) We would like to know how far each of the runners will have run when they pass the starting line at the same time. We first find LCM(75, 90), which will tell us how much time has elapsed when they pass the starting line at the same time. The prime factorizations of 75 and 90 are 75 = and 90 = These share a factor of 3 and a factor of 5, so the diagram looks like Factors of 75 Factors of So LCM(75, 90) = 5 (3 5) (2 3) = 450. That is, 450 seconds after starting, the runners will pass each other at the starting line. The first runner will have run = 6 laps around the track, and the second will have run = 5 laps around the track. 11
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 informationSection 5.1 Number Theory: Prime & Composite Numbers
Section 5.1 Number Theory: Prime & Composite Numbers Objectives 1. Determine divisibility. 2. Write the prime factorization of a composite number. 3. Find the greatest common divisor of two numbers. 4.
More informationMath 101 Study Session Quiz 1 Chapter 3 Sections 1 through 4
Math 101 Study Session Quiz 1 Chapter 3 Sections 1 through 4 July 28, 2016 Chapter 3 Section 1: The Least Common Multiple and Greatest Common Factor Natural number factors of a number divide that number
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 informationSieve of Erastosthenes: used to identify prime numbers Using the list of numbers from 1
Chapter 4 Number Theory ection 4.2, page 221 Prime and Composite Numbers Rectangle Dimensions Using the Dimensions of Rectangles Chart, use napcubes to create as many rectangles as possible using the
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 informationChapter 5 Number Theory. 5.1 Primes, composites, and tests for divisibility
Chapter 5 Number Theory 5.1 Primes, composites, and tests for divisibility Primes and Composites Prime number (prime) a counting number with exactly two different factors Composite number (composite) a
More information18. [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 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 informationGrade 5, Ch. 1 Math Vocabulary
Grade 5, Ch. 1 Math Vocabulary rectangular array number model fact family factors product factor pair divisible by divisibility rules prime number composite number square array square number exponent exponential
More informationmay be sent to:
B A S I C M A T H A SelfTutorial by Luis Anthony Ast Professional Mathematics Tutor LESSON 5: FACTORS, MULTIPLES & DIVISIBILITY Copyright 2005 All rights reserved. No part of this publication may be reproduced
More information54 Prime and Composite Numbers
54 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 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 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 informationUnit 1 Review Part 1 3 combined Handout KEY.notebook. September 26, 2013
Math 10c Unit 1 Factors, Powers and Radicals Key Concepts 1.1 Determine the prime factors of a whole number. 650 3910 1.2 Explain why the numbers 0 and 1 have no prime factors. 0 and 1 have no prime factors
More informationMath 10C: Numbers, Radicals, and Exponents PRACTICE EXAM
Math 10C: Numbers, Radicals, and Exponents PRACTICE EXAM 1. 1.273958... belongs to: The set of integers. The set of rationals. The set of irrationals. None of the above. 2. 7.4 belongs to: The set of integers.
More informationMEP Y8 Practice Book A
2 Factors MEP Y8 Practice Book A 2.1 Factors and Prime Numbers A factor divides exactly into a number, leaving no remainder. For example, 13 is a factor of 26 because 26 13 = 2 leaving no remainder. A
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 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 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 informationFactoring Numbers. Factoring numbers means that we break numbers down into the other whole numbers that multiply
Factoring Numbers Author/Creation: Pamela Dorr, September 2010. Summary: Describes two methods to help students determine the factors of a number. Learning Objectives: To define prime number and composite
More informationHow To Math Properties
CLOSURE a + b is a real number; when you add 2 real numbers, the result is also a real number. and 5 are both real numbers, + 5 8 and the sum, 8, is also a real number. a b is a real number; when you subtract
More informationClass VI Chapter 3 Playing with Numbers Maths
Exercise 3. Question : Write all the factors of the following numbers: (a) 24 (b) 5 (c) 2 (d) 27 (e) 2 (f) 20 (g) 8 (h) 23 (i) 36 (a) 24 24 = 24 24 = 2 2 24 = 3 8 24 = 4 6 24 = 6 4 Factors of 24 are, 2,
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 informationGCF/ Factor by Grouping (Student notes)
GCF/ Factor by Grouping (Student notes) Factoring is to write an expression as a product of factors. For example, we can write 10 as (5)(2), where 5 and 2 are called factors of 10. We can also do this
More informationBlack GCF and Equivalent Factorization
Black GCF and Equivalent Factorization Here is a set of mysteries that will help you sharpen your thinking skills. In each exercise, use the clues to discover the identity of the mystery fraction. 1. My
More informationBecause 6 divides into 50 eight times with remainder 2, 6 is not a factor of 50.
CHAPTER 3 FACTORS AND MULTIPLES Factors and multiples deal with dividing and multiplying positive integers1,2,3,4,. In this chapter you will work with such concepts as Greatest Common Factor (GCF) and
More informationMTH 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 informationMAT Mathematical Concepts and Applications
MAT.1180  Mathematical Concepts and Applications Chapter (Aug, 7) Number Theory: Prime and Composite Numbers. The set of Natural numbers, aka, Counting numbers, denoted by N, is N = {1,,, 4,, 6,...} If
More informationCOMPASS Numerical Skills/PreAlgebra Preparation Guide. Introduction Operations with Integers Absolute Value of Numbers 13
COMPASS Numerical Skills/PreAlgebra 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 informationFinding Prime Factors
Section 3.2 PREACTIVITY PREPARATION Finding Prime Factors Note: While this section on fi nding prime factors does not include fraction notation, it does address an intermediate and necessary concept to
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 informationLESSON 4 Missing Numbers in Multiplication Missing Numbers in Division LESSON 5 Order of Operations, Part 1 LESSON 6 Fractional Parts LESSON 7 Lines,
Saxon Math 7/6 Class Description: Saxon mathematics is based on the principle of developing math skills incrementally and reviewing past skills daily. It also incorporates regular and cumulative assessments.
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 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 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 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 informationIntroduction 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 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. There are two semi trucks that come past my house. The first one comes past every 80
Name Hour  LCM and GCF Quiz Please solve each question and show your work. Make sure that your answer is in WORDS and answers the question being asked. 1. There are two semi trucks
More informationClass Overview: We have finished with greatest common factor (GCF). The students will spend the next 3 days on least common multiple (LCM)
Group Lesson Plan Assignment Ranetta Goss & Santhi Prabahar Title of Lesson: Introducing Least Common Multiples (LCM) Topic: LCM Grade Level: 6 Georgia Performance Standards: M6N1. Students will understand
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 informationPrime and Composite Numbers Prime Factorization
Prime and Composite Numbers Prime Factorization Reteaching Math Course, Lesson A prime number is a whole number greater than that has exactly two factors, the number itself and. Examples: Factors of are
More informationM A T H E M A T I C S
M A T H E M A T I C S Grade 6 Mathematics Frameworks Unit 2 Fun and Games Student Edition TABLE OF CONTENTS Unit 2 FUN AND GAMES Overview...3 Enduring Understandings...3 Essential Questions...4 Key Standards
More informationNativity Catholic School Rising 7th grade IXL Language Arts and Math Summer Homework
Nativity Catholic School Rising 7th grade IXL Language Arts and Math Summer Homework Please work on the following skills listed in the 6th Grade Math and Language Arts IXL Program for a minimum of 60 minutes
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 82006 An Introduction to Number Theory Prime Numbers and Their Applications. Crystal
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 informationSCHOOL 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 informationSection 2.1/2.2 An Introduction to Number Theory/Integers. The counting numbers or natural numbers are N = {1, 2, 3, }.
Section 2.1/2.2 An Introduction to Number Theory/Integers The counting numbers or natural numbers are N = {1, 2, 3, }. A natural number n is called the product of the natural numbers a and b if a b = n.
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 information5.1 Text HW Number Theory Math 210 pbf8 1. Illustrate = 34 as in theorem 1.5 [i.e. showing that EVEN + EVEN = EVEN!].
.1 Text HW Number Theory Math 10 pbf8 1. Illustrate 1+18 = 4 as in theorem 1. [i.e. showing that EVEN + EVEN = EVEN!]. 1 18 4 11 48. Illustrate + 449 = 781 as in theorem 1.: EVEN + ODD = ODD 449 781 4a.
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 informationApplying Prime Factorization Grade Six
Ohio Standards Connection Number, Number Sense and Operations Standard Benchmark G Apply and explain the use of prime factorizations, common factors, and common multiples in problem situations. Indicator
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 informationThere 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 informationChapter 1 Basic Number Concepts
Draft of September 2014 Chapter 1 Basic Number Concepts 1.1. Introduction No problems in this section. 1.2. Factors and Multiples 1. Determine whether the following numbers are divisible by 3, 9, and 11:
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 10 M.K. HOME TUITION Mathematics Revision Guides Level: GCSE Foundation Tier FACTORS, PRIME NUMBERS, H.C.F. AND L.C.M. Version:
More informationExponents, Factors, and Fractions. Chapter 3
Exponents, Factors, and Fractions Chapter 3 Exponents and Order of Operations Lesson 31 Terms An exponent tells you how many times a number is used as a factor A base is the number that is multiplied
More information1.3 Polynomials and Factoring
1.3 Polynomials and Factoring Polynomials Constant: a number, such as 5 or 27 Variable: a letter or symbol that represents a value. Term: a constant, variable, or the product or a constant and variable.
More information3 PLAYING WITH NUMBERS
3 PLAYING WITH NUMBERS Exercise 3.1 Q.1. Write all the factors of the following numbers : (a) 24 (b) 15 (c) 21 (d) 27 (e) 12 (f) 20 (g) 18 (h) 23 (i) 36 Ans. (a) 24 = 1 24 = 2 12 = 3 8 = 4 6 Hence, factors
More informationMiddle School Math Assignment: How to Access IXL on a Computer or ipad
Middle School Math Assignment: How to Access IXL on a Computer or ipad Accessing IXL on a Computer: To log in to IXL you should go to the following site: www.ixl.com/signin/stmeshouston You will see this
More informationObjective. Materials. find a relationship between Greatest Common Factor (GCF) and Least Common Multiple (LCM) justify why Oliver s method works
. Objective To discover a relationship between the Greatest Common Factor (GCF) and the Least Common Multiple (LCM) of two numbers A c t i v i t y 4 Oliver s Method Materials TI73 calculator Student Worksheet
More informationWhen multiplying whole numbers we are counting a repeated set of items. Exercise 1: How many pizza boxes are there?
When multiplying whole numbers we are counting a repeated set of items. Exercise 1: How many pizza boxes are there? 1 2 3 1 2 3 4 There are 3 rows and 4 columns of boxes. Thus, we have 3 x 4 = 12 pizza
More informationHomework 5 Solutions
Homework 5 Solutions 4.2: 2: a. 321 = 256 + 64 + 1 = (01000001) 2 b. 1023 = 512 + 256 + 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1 = (1111111111) 2. Note that this is 1 less than the next power of 2, 1024, which
More information51 NUMBER THEORY: DIVISIBILITY; PRIME & COMPOSITE NUMBERS 210 f8
51 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 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 informationStandardsBased Progress Mathematics. Progress in Mathematics
SADLIER StandardsBased Progress Mathematics Aligned to SADLIER Progress in Mathematics Grade 5 Contents Chapter 1 Place Value, Addition, and Subtraction......... 2 Chapter 2 Multiplication....................................
More informationCategory 3 Number Theory Meet #1, October, 2000
Category 3 Meet #1, October, 2000 1. For how many positive integral values of n will 168 n be a whole number? 2. What is the greatest integer that will always divide the product of four consecutive integers?
More informationMath 1111 Journal Entries Unit I (Sections , )
Math 1111 Journal Entries Unit I (Sections 1.11.2, 1.41.6) Name Respond to each item, giving sufficient detail. You may handwrite your responses with neat penmanship. Your portfolio should be a collection
More informationGRADES 7, 8, AND 9 BIG IDEAS
Table 1: Strand A: BIG IDEAS: MATH: NUMBER Introduce perfect squares, square roots, and all applications Introduce rational numbers (positive and negative) Introduce the meaning of negative exponents for
More informationZero: If P is a polynomial and if c is a number such that P (c) = 0 then c is a zero of P.
MATH 11011 FINDING REAL ZEROS KSU OF A POLYNOMIAL Definitions: Polynomial: is a function of the form P (x) = a n x n + a n 1 x n 1 + + a x + a 1 x + a 0. The numbers a n, a n 1,..., a 1, a 0 are called
More informationAlgebra 1 Course Title
Algebra 1 Course Title Course wide 1. What patterns and methods are being used? Course wide 1. Students will be adept at solving and graphing linear and quadratic equations 2. Students will be adept
More informationSQUARE ROOTS AND CUBE ROOTS
9 SQUARE ROOTS AND CUBE ROOTS. Find the smallest natural number by which the following numbers must be multiplied to make them a perfect square: (i) 68 (ii) 8 Ans. (i) 68 9 9 Since the factor does not
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 informationAlgebra I. Copyright 2014 Fuel Education LLC. All rights reserved.
Algebra I COURSE DESCRIPTION: The purpose of this course is to allow the student to gain mastery in working with and evaluating mathematical expressions, equations, graphs, and other topics, with an emphasis
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 informationNumber. ch?v=mquhqkknldk (maths is confusing funny)
Number http://www.youtube.com/watch?v =52CzD31SqaM&feature=related (maths is confusing II funny) http://www.youtube.com/wat ch?v=mquhqkknldk (maths is confusing funny) SLO To find multiples of a number
More informationWhen factoring, we look for greatest common factor of each term and reverse the distributive property and take out the GCF.
Factoring: reversing the distributive property. The distributive property allows us to do the following: When factoring, we look for greatest common factor of each term and reverse the distributive property
More informationGrade 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 informationUNIT 2 VOCABULARY: DIVISIBILITY
1º ESO Bilingüe Página 1 UNIT 2 VOCABULARY: DIVISIBILITY 1.1. Multiples and factors We say that a number b is a factor or a divisor of another number a if the division a : b is an exact division. If a
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 informationTips, 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 information5544 = 2 2772 = 2 2 1386 = 2 2 2 693. Now we have to find a divisor of 693. We can try 3, and 693 = 3 231,and we keep dividing by 3 to get: 1
MATH 13150: Freshman Seminar Unit 8 1. Prime numbers 1.1. Primes. A number bigger than 1 is called prime if its only divisors are 1 and itself. For example, 3 is prime because the only numbers dividing
More informationALGEBRA 1/ALGEBRA 1 HONORS
ALGEBRA 1/ALGEBRA 1 HONORS CREDIT HOURS: 1.0 COURSE LENGTH: 2 Semesters COURSE DESCRIPTION The purpose of this course is to allow the student to gain mastery in working with and evaluating mathematical
More informationMultiplying Polynomials 5
Name: Date: Start Time : End Time : Multiplying Polynomials 5 (WS#A10436) Polynomials are expressions that consist of two or more monomials. Polynomials can be multiplied together using the distributive
More informationAlgebra for Digital Communication
EPFL  Section de Mathématiques Algebra for Digital Communication Fall semester 2008 Solutions for exercise sheet 1 Exercise 1. i) We will do a proof by contradiction. Suppose 2 a 2 but 2 a. We will obtain
More informationMATH 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 informationPrime Time Notes. Problem 1.1
Problem 1.1 factor  one of two or more whole numbers that when multiplied together give a product proper factor  all the factors of that number, except the number itself abundant numbers  numbers whose
More information5.1 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 informationUnit 2 Chapter 2 Section 2 Simplifying Expressions.notebook. October 22, Bellringer. Jul 26 12:38 PM. Jun 6 7:28 PM.
Bellringer 1. Get a calculator. Write down tonight's HW. 3. Take out last night's HW Unit packet p. 15 4. Write a variable expression that represents each phrase. Check Your Homework  p.15 a) Eleven more
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 informationAlgebra 1A and 1B Summer Packet
Algebra 1A and 1B Summer Packet Name: Calculators are not allowed on the summer math packet. This packet is due the first week of school and will be counted as a grade. You will also be tested over the
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 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 informationFactoring (pp. 1 of 4)
Factoring (pp. 1 of 4) Algebra Review Try these items from middle school math. A) What numbers are the factors of 4? B) Write down the prime factorization of 7. C) 6 Simplify 48 using the greatest common
More informationName Date Block. Algebra 1 Laws of Exponents/Polynomials Test STUDY GUIDE
Name Date Block Know how to Algebra 1 Laws of Eponents/Polynomials Test STUDY GUIDE Evaluate epressions with eponents using the laws of eponents: o a m a n = a m+n : Add eponents when multiplying powers
More informationFun Number Theory Meeting (Computation, Factors, Exponent Patterns, Factorials, Perfect Squares and Cubes)
Fun Number Theory Meeting (Computation, Factors, Exponent Patterns, Factorials, Perfect Squares and Cubes) Topic This meeting s topics fall in the category of number theory. Students will apply their knowledge
More informationRESOURCE FOR THE COURSE 1 CARNEGIE TEXTBOOK. SECTION and PAGE NUMBER TOPIC. 1.1 Pages 514 Factors and Multiples
RESOURCE FOR THE COURSE 1 CARNEGIE TEXTBOOK SECTION and PAGE NUMBER TOPIC 1.1 Pages 514 Factors and Multiples Properties of Addition and Multiplication (Commutative, Associative and Distributive) Apply
More informationNAME 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 information15 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