# Prime Factorization - Mrs. Dixon's Website

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Page 1 of 5Prime FactorizationBEFORENowWHY?You multiplied whole numbersto find their product.You’ll write a number as aproduct of prime numbers.So you can tell whether a ChineseNew Year is prime, as in Ex. 51.You can make a list of prime numbers.Word Watchprime number, p. 157composite number, p. 157prime factorization, p. 158factor tree, p. 1581 Write the whole numbers from 2 through 50.2 Cross out all multiples of 2 other than 2. (The first row in the list below has beendone for you.) Then go to the next remaining number after 2 and cross out all itsmultiples other than itself. Repeat until you can no longer cross out 272829303132333435363738394041424344454647484950In the activity, the numbers that are not crossed out are called primenumbers. A prime number is a whole number greater than 1 whoseonly whole number factors are 1 and itself. A composite number is awhole number greater than 1 that is not prime. For example, 5 is a primenumber while 6 2 3 is a composite number. The number 1 is neitherprime nor composite.EXAMPLE1Writing Factors of a NumberField Trip A class of 36 students is on a field trip at the zoo. The teacherwants to break the class into groups of the same size. Find all thepossible group sizes by writing all the factors of 36.Solution36 1 36 2 18 3 12Poison dart frog froma zoo in New York City 4 936 isn’t divisible by 5. Skip to 6. 6 636 isn’t divisible by 7 and 8. Skip to 9. 9 4Stop when the factors repeat.ANSWER The possible group sizes are 1, 2, 3, 4, 6, 9, 12, 18, and 36.Lesson 4.1Prime Factorization157

Page 2 of 5EXAMPLEwith2Identifying Prime and Composite NumbersTell whether the number is prime or composite.ReviewNeed help with divisibilityrules? See p. 685.a. 56b. 11Solutiona. The factors of 56 are 1, 2, 4,7, 8, 14, 28, and 56. So, 56is composite.b. The only factors of 11 are1 and 11. So, 11 is prime.Prime Factorization Expressing a whole number as a product of primenumbers is called prime factorization . You can use a diagram called afactor tree to write the prime factorization of a number. Use an exponentwhen a prime factor appears more than once in the prime factorization.EXAMPLEwithReading3Using a Factor TreeUse a factor tree to write the prime factorization of 54.To factor a number meansto write the number as aproduct of its factors.One possible factor tree:54Write original number.2 27Factor 54 as 2 27.2 3 9Factor 27 as 3 9.2 3 3 3Factor 9 as 3 3.Another possible factor tree:54Write original number.3 18Factor 54 as 3 18.3 3 6Factor 18 as 3 6.3 3 2 3Factor 6 as 2 3.Both factor trees give the same result: 54 2 3 3 3 2 33.ANSWER The prime factorization of 54 is 2 33.Your turn nowUse a factor tree to write the prime factorizationof the number.1. 30158Chapter 4Number Patterns and Fractions2. 483. 444. 75

Page 3 of 5INTERNETExerciseseWorkbook PlusCLASSZONE.COMMore Practice, p. 708Getting Ready to Practice1. Vocabulary In your own words, describe the difference between aprime number and a composite number.Tell whether the number is prime or composite. Explain your reasoning.tch Out!Wa2. 53. 104. 155. 436. 227. Guided Problem Solving You are a tour guide and want to divideA number may bedivisible by the same primenumber multiple times.90 people into the same size tour groups. The ideal tour group sizeis between 11 and 15 people. How many people should be in eachtour group?1 Find all the factors of 90.2 Use the factors of 90 to find all the possible group sizes.3 Is more than one answer possible? Explain your reasoning.Practice and Problem SolvingWrite all the factors of the number.withExample1238. 209. 4510. 2411. 1312. 2113. 1814. 1615. 5416. 10017. 60HomeworkTell whether the number is prime or composite. Explain your reasoning.Exercises8–17, 5218–27, 5132–4418. 8819. 2320. 6121. 3922. 5123. 6724. 4125. 9926. 8727. 201Online ResourcesCLASSZONE.COM More Examples eTutorial Plus28. List the first 10 prime numbers.29. Critical Thinking What is the only even prime number?30. Souvenir Pouches As a volunteer at a museum, youfill souvenir pouches with semi-precious stones.Each pouch has the same number of stones,and there are no leftover stones. Is thetotal number of stones in thesouvenir pouches primeor composite? Explainyour reasoning.Lesson 4.1Prime Factorization159

Page 4 of 531. Find the Error36Describe and correctthe error in writing theprime factorizationof 36.2 182 2 9The prime factorizationof 36 is 22 9.Copy and complete the factor tree. Then write the prime factorization.32.33.1234.521102 ?2 ?11 ? 2 3? 2 ?11 ? ?Use a factor tree to write the prime factorization of the number.35. 2636. 4937. 6838. 5039. 6440. 14441. 22542. 58843. 61244. 86445. Writing Explain the difference between finding the factors of anumber and finding the prime factorization of a number.HistoryAlgebra Tell whether the value of the variable expression is primeor composite.46. 6p 2 when p 547. 3x 1 when x 6249. r 3 17 when r 448. 7n 3 when n 250. Goldbach’s Conjecture A conjecture is a statement believed to be truebut not proved to be true. Christian Goldbach (1690–1764) made thisconjecture about prime numbers: Every even number greater than2 can be written as the sum of two prime numbers. Show thatGoldbach’s conjecture is true for every even number between 3 and 11.51. Chinese New Year The year 2019 is the Year of the Pig. Is 2019prime or composite? Explain your reasoning.52. Desks in a Classroom A classroom in your school contains 32 desks, and another classroom contains 35 desks. Which classroom allows formore rectangular desk arrangements if you use all the desks?Explain your answer.Chinese New YearThe Chinese New Year,which falls between lateJanuary and earlyFebruary, is associatedwith an animal. Twelveanimals are repeated in12-year cycles. The year2019 is the Year of thePig. What year will bethe next Year of the Pig?160Chapter 4Use a factor tree to write the prime factorization of the number.53. 176454. 108955. 123258. Challenge Identify whichcolumns of the table mostlycontain composite numbers.Explain why.Number Patterns and Fractions7131956. 231028142039152157. 2205410162251117236121824

Page 5 of 5Mixed ReviewFind the mean, median, mode(s), and range of the data. (Lesson 3.1)59. 4, 6, 4, 7, 8, 3, 9, 4, 3, 260. 12, 15, 14, 20, 25, 13, 18, 19, 8Choose a Strategy Use a strategy fromthe list to solve the following problem.Explain your choice of strategy.61. A family of 2 adults and 2 children buysProblem Solving StrategiesLook for a PatternEstimate Make a Table a family season pass to a water park for 249.99. A single adult ticket costs 39.99, and a single child ticketcosts 27.99. How many times must the entire family go to thewater park for the pass to be worth its cost?Basic Skills Test the number for divisibility by 2, 3, 5, 6, 9,and 10.62. 14463. 34564. 204065. 2514Test-Taking Practice66. Short Response Give two different factor trees for 348.INTERNETState Test Practice67. Multiple Choice What is the prime factorization of 72?CLASSZONE.COMA. 22 3 6B. 3 3 22C. 2 3 32D. 2 3 9What am I?If a number in the list below is a factor of 2,343,750,write its corresponding letter on a piece of paper.Unscramble the letters to answer the riddle.2C3A4E5H6M7L8N9RRiddle:Take me out and scratch my head,I am now black, but once was red.What am I?Answer:I am a(n) ? .10TLesson 4.1Prime Factorization161

Use a factor tree to write the prime factorization of the number. 1. 30 2. 48 3. 44 4. 75 Your turn now Prime FactorizationExpressing a whole number as a product of prime numbers is called . You can use a diagram called a to write the prime factorization of a number. Use an exponent when a prime factor

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