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1.2 Prime Factorization GOAL Express a composite number as the product of prime factors. Learn about the Math Teo and Sheree need passwords for their e-mail accounts. They want to use numbers that they can remember, but will be difficult for others to figure out. Sheree uses the prime factorization of her address, 1050, to form part of her password. Teo uses the prime factorization of his date of birth, 330 (March 30). 4 5 1050 11 18 25 1 6 12 19 26 2 7 13 20 27 3 8 9 14 15 21 28 22 29 16 23 30 prime factorization the representation of a composite number as the product of its prime factors; for example, the prime factorization of 24 is 24 2 2 2 3, or 23 3; usually, the prime numbers are written in order from least to greatest 10 17 24 31 ? What are Teo’s and Sheree’s passwords? Example 1: Using a factor tree to determine prime factors If Sheree’s address is 1050, what will her password be? Sheree’s Solution 1050 10 5 105 2 5 21 7 3 1050 2 3 5 5 7 2 3 52 7 My e-mail password will be my address followed by its prime factors: 105023557. 8 Chapter 1 I know that 10 is a factor of 1050. I divided by 10 to determine another factor, 105. I used a factor tree to show these two factors. 5 is a factor of both 10 and 105. I divided each number by 5 to determine two more factors, 2 and 21. I continued to calculate factors until only prime factors were left at the ends of the branches. I wrote the prime numbers in order, from least to greatest, to create the prime factorization of 1050. Then I used exponents to write the prime factorization a different way. NEL Example 2: Using repeated division to determine prime factors If Teo’s date of birth is 330, what will his password be? Teo’s Solution 165 23 3 0 33 51 6 5 23 3 0 11 33 3 51 6 5 23 3 0 330 2 3 5 11 My e-mail password will be my date of birth followed by its prime factors: 33023511. I divided by prime factors. The prime number 2 is a factor of 330. So, I divided by 2 to determine another factor, 165. The prime number 5 is a factor of 165. I divided by 5 to determine another factor, 33. I divided by 3 to get the last prime factor. I wrote the prime factors as a multiplication sentence to show the prime factorization of 330. Reflecting 1. What might Sheree’s factor tree have looked like if she had started dividing by 50 instead of 10? 2. Does the order in which you divide by factors change the prime factorization? Provide an example to support your answer. 3. a) What divisibility rules might Sheree have used to determine some factors of 1050, without dividing the number? b) What divisibility rules might Teo have used to determine some factors of 330, without dividing the number? 4. If you used Sheree’s strategy and Teo’s strategy for the same number, would you get the same prime factorization? Explain. Use an example to support your explanation. NEL Number Relationships 9 Work with the Math Example 3: Writing the prime factorization of a composite number What is the prime factorization of 1470? Solution A: Creating a factor tree Solution B: Using repeated division 1470 10 5 147 2 3 49 7 7 Write the prime numbers at the ends of the branches in order, from least to greatest, to show the prime factorization. 1470 2 3 5 7 7 2 3 5 72 A 3 72 1 71 4 7 57 3 5 21 4 7 0 Write the prime numbers in the divisors and quotient in order, from least to greatest, to show the prime factorization. 1470 2 3 5 7 7 2 3 5 72 8. Factor trees are being used to determine the prime factorizations of 1755 and 2180. Checking 5. Determine the prime factorization of each number. a) 117 b) 147 c) 220 d) 270 6. a) Rivka used the last four digits of her telephone number, 1048, followed by its prime factorization to create her e-mail password. Determine her password. b) Identify any divisibility rules you used. B Practising 7. Determine the prime factorization of each number. a) b) c) d) 10 100 102 320 375 Chapter 1 e) f) g) h) 412 2055 512 3675 1755 2180 3 585 10 218 a) Explain how you know that each factor tree is not complete. b) Copy and complete each factor tree to determine the prime factorization. 9. Determine the missing number in each prime factorization. a) b) c) d) 200 2 2 ■ 5 5 216 23 3■ 8281 7 7 13 ■ 1568 ■ 5 72 10. a) Determine the prime factorization of each number. 64 256 1024 b) What does the prime factorization tell you about each number? NEL 11. a) Determine the prime factorization of a three-digit or four-digit composite number of your choice. b) Create an e-mail password using the composite number you chose, followed by its prime factorization. 12. Why would you determine the prime factorization of only composite numbers? Use an example to support your explanation. 13. How can you use the prime factorization of a number to determine whether the number is even or odd? Use an example to support your explanation. C Extending 14. The prime factorization of a number is 23 5 7. a) Explain how you can use the prime factorization to determine whether 35 is a factor of the number. b) Explain how you can use the prime factorization to determine whether 8 is a factor of the number. c) How do you know, without multiplying the prime factors, that the last digit of the number is 0? 15. The prime factorizations of two numbers are shown. 25 3 52 710 38 5 7 4112 a) How do you know that 2 is not a common factor of the numbers? b) How do you know that 35 is a common factor? c) List another common factor of the numbers. Explain your reasoning. NEL 16. a) Multiply any three-digit number by 1001 to get a six-digit number. 1001 ■■■ ———— b) Divide your six-digit number by 7, 11, and then 13. What is the quotient? c) Show the prime factorization of 1001. d) Explain how you could use the prime factorization of 1001 to predict the quotient you calculated in part (b). 17. a) Multiply two different prime numbers. List all the possible factors of the product. b) Repeat part (a) with two other prime numbers. c) What do you notice about the number of factors each product has? 18. a) Multiply three different prime numbers. List all the possible factors of the product. b) Repeat part (a) with three other prime numbers. c) What do you notice about the number of factors each product has? 19. Use your results in questions 17 and 18 to predict the number of factors the product of four different prime numbers will have. Use an example to check your prediction. 20. The prime factorization of a number is 25 38 57 74 11 13. Which statements are true about the number? Explain your reasoning. a) b) c) d) e) The number is even. The number is a multiple of 10. 15 is a factor of the number. 17 is not a factor of the number. 77 is a factor of the number. Number Relationships 11