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3.1 Factors and Multiples of Whole Numbers

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A prime number A Composite number A factor of a Number Prime Factorization Greatest Common Factor Common Multiple NEW VOCABULARY

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A Prime Number Is exactly divisible only by itself and 1 The number 1 is not considered a prime number Examples: 2, 3, 5, 7, 11, 13, 17, 19...

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have factors and can be written as the product of prime numbers Example: 12 = 2 x 2 x 3 = 2² x 3 A Composite Number This multiplying is called PRIME FACTORING

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The factoring of numbers: can only be done in one way Example: 6 = 2 x 3 or 3 x 2 Example: 6 = 1 x 6 or 6 x 1 Prime Factorization No other number can be written as the product of prime factors in this way This is not prime factoring because neither 6 nor 1 are Prime Numbers

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Determining the Prime Factors of a Whole Number: Write the Prime Factorization of 3300 Method #1 (Factor Tree): Draw a factor tree. Write 3300 as a product of 2 factors. Both 33 and 100 are composite numbers, so we can factor again. Both 3 and 11 are prime factors, but 4 and 25 can be factored further. The prime factors of 3300 are 2, 3, 5, and 11. The prime factorization of 3300 is: 2 · 2 · 3 · 5 · 5 · 11, or 2 2 · 3 · 5 2 · 11

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Determining the Prime Factors of a Whole Number: Write the Prime Factorization of 3300 Method #2 (Repeated Division): Use Repeated Division by prime factors. Begin by dividing 3300 by the least prime factor, which is 2. Divide by 2 until 2 is no longer a factor. Continue to divide each quotient (= result) by a prime factor until the quotient is 1. 3300 : 2 = 1650 1650 : 2 = 825 825 : 3 = 275 275 : 5 = 11 11 : 11 = 1

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POWERPOINT PRACTICE PROBLEM Write the Prime Factorization of 2646

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The Greatest Common Factor (GCF) Is the greatest factor the numbers have in common

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What is the GCF of 12 and 20 Method #2 (Repeated Division):Method #1 (Factor Tree): 3300 : 2 = 1650 1650 : 2 = 825 825 : 3 = 275 275 : 5 = 11 11 : 11 = 1

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What is the GCF of 12 and 20 Method #2 (Repeated Division): Use Repeated Division to determine all the Prime Factors of each number Write Prime Factorization for both of them Pick the which one(s) are common to both and multiply them 12 : 2 = 6 6 : 2 = 3 3 : 3 = 1 Prime Factors 20 : 2 = 10 10 : 2 = 5 5 : 5 = 1 Prime Factors

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POWERPOINT PRACTICE PROBLEM What is the GCF of 126 and 144

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The Least Common Multiple (LCM) Of two or more numbers is the least number that is divisible by each number.

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Determining the LCM of 18, 20, and 30 Method #2 (Repeated Division): Use Repeated Division to determine all the Prime Factors of each number Write Prime Factorization for each of them Identify all the different factors in all the numbers. Highlight the greatest power of each prime factor in any list. 18 : 2 = 9 9 : 3 = 3 3: 3 = 1 20 : 2 = 10 10 : 2 = 5 5: 5 = 1 30 : 2 = 15 15 : 3 = 5 5: 5 = 1 All different factors = 2, 3, 5 The greatest power of these factors = 2 2, 3 2, 5

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Determining the LCM of 18, 20, and 30 Method #2 (Repeated Division): Use Repeated Division to determine all the Prime Factors of each number Write Prime Factorization for each of them Identify all the different factors in all the numbers. Highlight the greatest power of each prime factor in any list. Multiply them 18 : 2 = 9 9 : 3 = 3 3: 3 = 1 20 : 2 = 10 10 : 2 = 5 5: 5 = 1 30 : 2 = 15 15 : 3 = 5 5: 5 = 1 Multiply these factors = 2 2 ∙ 3 2 ∙ 5 = 180 The greatest power of these factors = 2 2, 3 2, 5

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POWERPOINT PRACTICE PROBLEM What is the LCM of 28, 42, and 63

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Where do we use Prime Factoring, GCF, and LCM?

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It helps us simplify fractions, radicals, and factor algebraic expressions 42 54 √64

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HOMEWORK O PAGES: 140 – 141 O PROBLEMS: 3 – 6 (a, d, f), 8 – 11 (a,b,d), 15 – 16 (a,b,d), 21

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