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Prime Time Problem 4.1 Answers A.)Answers will vary, but the longest string in the puzzle is: 2 x 2 x 2 x 3 x 5 x 7. B.)It is not possible to find a.

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Presentation on theme: "Prime Time Problem 4.1 Answers A.)Answers will vary, but the longest string in the puzzle is: 2 x 2 x 2 x 3 x 5 x 7. B.)It is not possible to find a."— Presentation transcript:

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3 Prime Time Problem 4.1 Answers A.)Answers will vary, but the longest string in the puzzle is: 2 x 2 x 2 x 3 x 5 x 7. B.)It is not possible to find a longer string than is in the puzzle. The longest possible string is 2 x 2 x 2 x 3 x 5 x 7. C.)Some strings can be broken down further to get longer strings. For example 420 x 2 can be split up into 210 x 2 x 2. D.)If all the numbers in a string are prime, the string is the longest possible. E.)Except for order, every whole number has exactly one longest string.

4 How many ways can you factor 100? Let's do it together.

5 Investigation 4 Notes VO CABULARY: Fundamental Theorem of Arithmetic - a whole number can be ________, except for order, into a _____________ of ___________ in exactly ____ way. for example, 100 can be written as: ______________________ factorization - a string of ________________ for example, one factorization for 100 is: ______________________ prime factorization - factor string of ________ numbers for example, the prime factorization for 100 is: ___________________ factor tree - an orderly record of your steps to find __________ _______________ Ex ***Although you arrive at the final product in different ways, all have the same prime factorization. Except for order, there is only ONE way to write it. exponents: small raised numbers used to tell how many times a _________ is repeated. For example, the prime factorization for 100 can be written using exponents as follows: ________________________ (this is called the short-cut method) Shortcuts to finding the GCF and LCM Greatest Common Factor (GCF) - The product of the longest prime factorization string that both numbers have in common. For ex. 24 = 2 x 2 x 2 x 360 = 2 x 2 x 3 x 5 GCF = 2 x 2 x 3 = 12 Use this method to find the GCF of 125 and 80 Least Common Multiple (LCM) - The product of the shortest prime factorization string that both numbers have in common. 24 = 2 x 2 x 2 x 360 = 2 x 2 x 3 x 5 LCM = 2 x 2 x 3 x 2 x 5 = 120 Use this method to find the LCM of 125 and 80

6 Investigation 4 Notes - Answer VO CABULARY: Fundamental Theorem of Arithmetic - a whole number can be factored, except for order, into a product of primes in exactly one way. for example, 100 can be written as: 2 x 2 x 5 x 5 factorization - a string of factors for example, one factorization for 100 is: 2 x 25 x 2 prime factorization - factor string of prime numbers for example, the prime factorization for 100 is: 2 x 2 x 5 x 5 factor tree - an orderly record of your steps to find prime factorization Ex ***Although you arrive at the final product in different ways, all have the same prime factorization. Except for order, there is only ONE way to write it. exponents: small raised numbers used to tell how many times a factor is repeated. For example, the prime factorization for 100 can be written using exponents as follows: 2² x 5² (this is called the short-cut method (using exponential notation) Shortcuts to finding the GCF and LCM Greatest Common Factor (GCF) - The product of the longest prime factorization string that both numbers have in common. For ex. 24 = 2 x 2 x 2 x 360 = 2 x 2 x 3 x 5 GCF = 2 x 2 x 3 = 12 Use this method to find the GCF of 125 and = 5 x 5 x 5 80 = 5 x 2 x 2 x 2 GCF = 5 Least Common Multiple (LCM) - The product of the shortest prime factorization string that both numbers have in common. 24 = 2 x 2 x 2 x 360 = 2 x 2 x 3 x 5 LCM = 2 x 2 x 3 x 2 x 5 = 120 Use this method to find the LCM of 125 and = 5 x 5 x 5 80 = 5 x 2 x 2 x 2 LCM = 5 x 5 x 5 x 2 x 2 x 2 = 1,000

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14 Prime Time Problem 4.2 Answers A.)From the pictures that we draw, we can read that 100 = 2 x 2 x 5 x 5. This string is a factorization of 100 into prime numbers. We now know (from Problem 4.1) that there is only 1 prime factorization for a number. Therefore, we call it the factorization instead of a factorization. B.)72 = 2 x 2 x 2 x 3 x = 2 x 2 x 2 x 3 x = 2 x 2 x 2 x 3 x 5 x 5 C.)72 = 2³x 3² 120 = 2³x 3 x = 2³x 3 x 5² D1.) Answer may vary - example will use = 3 x 3 x 2 x 2 x 2 D2.) You could circle the other factors left in the factorization. 72 = 3 x 3 x 2 x 2 x 2 2 x 2 x 2 = 8 so 9 is paired with 8. E.)Many possible answers. Example will use the multiple 144. Prime factorization of 72 = 3 x 3 x 2 x 2 x 2 Prime factorization of 144 = 3 x 3 x 2 x 2 x 2 x 2 The only difference is another factor of 2 is included because 144 is the second multiple of 72. Accordingly, every prime factorization for a multiple of 72 will contain the prime factorization of 72 in it.

15 Pull Way to Go! Make a Prime Factorization Tree 30 Hel p The last number on each arrow should be a prime number. Prime numbers are circled in black. Answer 2 x 15 3 x 5 2x3x5 Draw arrows and write the factors for the number until the factors are all prime numbers. Then Check your answers.

16 Pull Way to Go! Make a Prime Factorization Tree 84 Hel p The last number on each arrow should be a prime number. Prime numbers are circled in black. Answer 2 x 42 6 x 7 2 x 3 2x2x3x7 Draw arrows and write the factors for the number until the factors are all prime numbers. Then Check your answers.

17 Pull Way to Go! Make a Prime Factorization Tree 63 Hel p The last number on each arrow should be a prime number. Prime numbers are circled in black. Answer 3 x 21 3 x 7 3x3x7 Draw arrows and write the factors for the number until the factors are all prime numbers. Then Check your answers.

18 Pull Way to Go! Make a Prime Factorization Tree 91 Hel p The last number on each arrow should be a prime number. Prime numbers are circled in black. Answer 91 x 1(IS NOT a prime #) 91 is a prime number Draw arrows and write the factors for the number until the factors are all prime numbers. Then Check your answers.

19 Pull Way to Go! Make a Prime Factorization Tree 128 Hel p The last number on each arrow should be a prime number. Prime numbers are circled in black. Answer 2 x 64 8 x 8 2 x 4 2 x4 2 x 2 2 x 2 2x2x2x2x2x2x2 Draw arrows and write the factors for the number until the factors are all prime numbers. Then Check your answers.

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