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Prime Factorization

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**A composite number is a whole number that has more than two factors.**

A prime number is a whole number greater than 1 that has exactly two factors, 1 and itself. Three is a prime number because its only factors are 1 and 3. What are the rest of the prime numbers? A composite number is a whole number that has more than two factors. Six is a composite number because it has more than two factors—1, 2, 3, and 6. The number 1 has exactly one factor and is neither prime nor composite.

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**Tell whether each number is prime or composite.**

The factors of 11 are 1 and 11. The factors of 16 are 1, 2, 4, 8, and 16. 11 is prime. 16 is composite.

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**Tell whether each number is prime or composite.**

D. 7 The factors of 14 are 1, 2, 7, and 14. The factors of 7 are 1 and 7. 7 is prime. 14 is composite.

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**A composite number can be written as the product of its prime factors**

A composite number can be written as the product of its prime factors. This is called the prime factorization of the number. You can use a factor tree to find the prime factors of a composite number. You can write prime factorization by using exponents. The exponent tells how many times to use the base as a factor. Writing Math

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**24 8 · 3 4 · 2 2 · 2 Write the prime factorization of each number.**

Write 24 as the product of two factors. 8 · 3 4 · 2 Continue factoring until all factors are prime. 2 · 2 The prime factorization of 24 is 2 · 2 · 2 · 3 or 23 · 3.

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**150 30 · 5 10 · 3 2 · 5 Write the prime factorization of each number.**

Write 150 as the product of two factors. 30 · 5 Continue factoring until all factors are prime. 10 · 3 2 · 5 The prime factorization of 150 is 2 · 3 · 5 · 5, or 2 · 3 · 52.

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**36 18 · 2 9 · 2 3 · 3 Write the prime factorization of each number.**

Write 36 as the product of two factors. 18 · 2 9 · 2 Continue factoring until all factors are prime. 3 · 3 The prime factorization of 36 is 2 · 2 · 3 · 3 or 22 · 32.

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**90 45 · 2 9 · 5 3 · 3 Write the prime factorization of the number.**

Write 90 as the product of two factors. 45 · 2 9 · 5 Continue factoring until all factors are prime. 3 · 3 The prime factorization of 90 is 3 · 3 · 5 · 2, or 2 · 32 · 5.

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ASSIGNMENT WS 33 You are A-MAZE-ING!

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You can also use a step diagram to find the prime factorization of a number. At each step, divide by the smallest possible prime number. Continue dividing until the quotient is 1.

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**Repeated Division 40 2 20 2 2 10 5 5 1 2 x 2 x 2 x 5 = 2³ x 5**

Start by dividing by the smallest prime number. 40 2 20 2 Keep using 2 until it will not work anymore. 2 10 When you get to 1 you are finished 5 5 1 2 x 2 x 2 x 5 = 2³ x 5

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**Let’s try another one… 42 2 3 21 7 7 1 2 x 3 x 7**

Start by dividing by the smallest prime number. 42 2 3 2 will no longer work so you need to try the next prime number…3! 21 When you get to 1 you are finished 7 7 1 2 x 3 x 7

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Now you try! 2 18 2 28 3 9 2 14 3 3 7 7 1 1 2 x 32 22 x 7

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**Write the prime factorization of the number.**

325 Divide 325 by 5. Write the quotient below 325. 5 325 5 65 13 13 Stop when the quotient is 1. 1 The prime factorization of 325 is 5 · 5 · 13, or 52 · 13.

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**Write the prime factorization of the number.**

275 Divide 275 by 5. Write the quotient below 275. 5 275 5 55 11 11 Stop when the quotient is 1. 1 The prime factorization of 275 is 5 · 5 · 11, or 52 · 11.

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**Write the prime factorization of each number.**

476 Divide 476 by 2. Write the quotient below 476. 2 476 2 238 Keep dividing by a prime number. 7 119 17 17 1 Stop when the quotient is 1. The prime factorization of 476 is 2 · 2 · 7 · 17, or 22 · 7 · 17.

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**There is only one prime factorization for any given composite number**

There is only one prime factorization for any given composite number. The last example began by dividing 476 by 2, the smallest prime factor of 476. Beginning with any prime factor of 476 gives the same result. 2 476 7 476 2 238 2 68 7 119 2 34 17 17 17 17 1 1 The prime factorizations are 2 · 2 · 7 · 17 and 7 · 2 · 2 · 17, which are the same as 17 · 2 · 2 · 7.

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