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You will need • a calculator 1.3 Common Factors and Common Multiples GOAL Use prime factorization to identify common factors and common multiples. Learn about the Math Jordan and Reilly are creating a large square mural. The mural will be made of 36 cm by 48 cm rectangles covered with coloured squares. They want these squares to be as large as possible, measured in whole numbers of centimetres. 48 cm 36 cm Then Jordan and Reilly plan to arrange copies of the 36 cm by 48 cm rectangle to form a large square mural that measures the least possible whole number of centimetres. They decide to use the greatest common factor (GCF) and least common multiple (LCM) of 36 and 48 to determine the dimensions of both sizes of squares. are the dimensions of the small squares and the ? What large square mural? greatest common factor (GCF) the greatest whole number that divides into two or more other whole numbers with no remainder; for example, 4 is the greatest common factor of 8 and 12 least common multiple (LCM) the least whole number that has two or more given numbers as factors; for example, 12 is the least common multiple of 4 and 6 Example 1: Using a Venn diagram to identify the GCF What are the dimensions of the small squares? Use the greatest common factor (GCF) of 36 and 48. Jordan’s Solution 36 2 2 3 3 48 2 2 2 2 3 Prime factors of 36 3 Then I arranged the prime factors in a Venn diagram. The common prime factors are in the overlap. Prime 2 factors 2 of 48 2 3 2 First I wrote the prime factorization of 36 and 48. I multiplied the common prime factors to determine the other common factors of 36 and 48. So, 2, 3, 4 (2 2), 6 (2 3), and 12 (2 2 3) are the common factors of 36 and 48. The GCF is 2 2 3 12. A 12 cm by 12 cm square is the largest possible square that divides a 36 cm by 48 cm rectangle. 12 Chapter 1 NEL Example 2: Using a Venn diagram to identify the LCM What are the dimensions of the final square mural? Use the least common multiple (LCM) of 36 and 48. Reilly’s Solution 36 2 2 3 3 48 2 2 2 2 3 Prime factors of 36 3 2 Prime factors 2 of 48 Then I arranged the prime factors in a Venn diagram. I multiplied all the prime numbers in both circles to determine the LCM of 36 and 48. 2 3 First I wrote the prime factorization of 36 and 48. 2 144 cm 48 cm The LCM is 3 2 2 3 2 2 144. The final mural will be a 144 cm by 144 cm square. 36 cm 144 cm Reflecting 1. How did identifying the GCF and LCM help Jordan and Reilly decide on the dimensions of the small squares and the large square mural? 2. Why do you think Jordan multiplied the factors in the overlap of the Venn diagram to determine other common factors of 36 and 48? 3. Why do you think Reilly multiplied the numbers in the three sections of the Venn diagram to determine the LCM of 36 and 48? NEL Number Relationships 13 Work with the Math Example 3: Using Venn diagrams to identify the GCF and LCM Show how to use prime factorization to identify the GCF and LCM of each pair of numbers. a) 27 and 42 b) 18 and 35 Solution A Solution B Write the prime factorization of 27 and 42. 27 3 3 3 42 2 3 7 Write the prime factorization of 18 and 35. 18 2 3 3 35 5 7 Record the prime factors in a Venn diagram. Record the prime factors in a Venn diagram. Prime factors of 27 3 3 3 Prime factors of 42 2 7 The GCF is the product of the numbers in the overlap. The GCF of 27 and 42 is 3. The LCM is the product of the numbers in both circles. The LCM of 27 and 36 is 3 3 3 2 7 378. A Checking 4. Identify the GCF and LCM of each pair of numbers. a) 120 2 2 2 3 5 210 2 3 5 7 b) 252 2 2 3 3 7 60 2 2 3 5 5. a) Identify another common factor of each pair of numbers in question 4. b) Identify another common multiple of each pair of numbers in question 4. 14 Chapter 1 Prime factors of 18 3 2 3 Prime factors of 35 5 7 The GCF is the product of the numbers in the overlap. There are no prime factors in the overlap, but 1 is a common factor of both 18 and 35. So, the GCF of 18 and 35 is 1. The LCM is the product of the numbers in both circles. The LCM of 18 and 35 is 2 3 3 5 7 630. B Practising 6. Use prime factorization to identify at least three common factors and at least three common multiples of each pair of numbers. a) 48 and 60 b) 32 and 64 c) 24 and 32 d) 512 and 648 7. Identify the GCF and LCM of each pair of numbers. a) 78 2 3 13 442 2 13 17 b) 32 2 2 2 2 2 24 2 2 2 3 NEL 8. Use prime factorization to identify four common factors and four common multiples of each pair of numbers. a) 468 13 396 22 32 11 b) 840 23 3 5 7 2000 24 53 c) 1818 2 32 101 606 2 3 101 22 13. Given the GCF or LCM, what else do you know about each pair of numbers? a) b) c) d) 32 14. Explain how you can use these centimetre bars to identify the common factors and GCF of 6 and 8. 9. Identify the GCF and LCM of each pair of numbers. a) 64 and 240 b) 55 and 275 6 Prime Prime 5 factors factors of 360 2 2 of 480 2 2 3 3 2 15. Explain how to identify the GCF and LCM of a pair of numbers, if one number is a factor of the other number. 16. What pairs of numbers fit this description? List as many pairs as you can. “A pair of numbers has a sum of 100. One number is a multiple of 3. The other number is a multiple of 11.” b) Identify four other common factors. c) Identify four other common multiples. 11. A rectangle measures 72 cm by 108 cm. a) A 2 cm by 2 cm square can be used to cover the rectangle without any spaces or overlapping. Explain why. b) How can you use the common factors of 72 and 108 to identify all the squares that can be used to cover the rectangle? c) List all other squares with wholenumber dimensions that can be used to cover the rectangle. 12. Identify the GCF and LCM of each pair of numbers. NEL 8 c) 48 and 72 d) 120 and 200 10. a) Identify the GCF and LCM of 360 and 480. a) 40 and 48 b) 120 and 400 Two numbers have a GCF of 2. Two numbers have an LCM of 2. Two numbers have a GCF of 3. Two numbers have an LCM of 10. c) 101 and 200 d) 1024 and 1536 C Extending 17. The prime factorizations of two numbers, a and b, have some missing prime factors. a23■ b■■ a) The GCF of a and b is 5. What is the value of a? b) The LCM of a and b is 210. What is the value of b? 18. Show how you can use prime factorization and a Venn diagram to identify the GCF and LCM of 12, 48, and 64. Number Relationships 15