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CHAPTER 2 Fraction Notation: Multiplication and Division Slide 2Copyright 2011 Pearson Education, Inc. 2.1Factorizations 2.2Divisibility 2.3Fractions and Fraction Notation 2.4Multiplication and Applications 2.5Simplifying 2.6Multiplying, Simplifying, and Applications 2.7Division and Applications
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OBJECTIVES 2.1 Factorizations Slide 3Copyright 2011 Pearson Education, Inc. aDetermine whether one number is a factor of another, and find the factors of a number. bFind some multiples of a number, and determine whether a number is divisible by another. cGiven a number from 1 to 100, tell whether it is prime, composite, or neither. dFind the prime factorization of a composite number.
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2.1 Factorizations Factor Slide 4Copyright 2011 Pearson Education, Inc. In the product a b, a and b are factors. If we divide Q by d and get a remainder of 0, then the divisor d is a factor of the dividend Q.
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EXAMPLE Determine whether one number is a factor of another, and find the factors of a number. Not 0 The remainder is not 0, so 12 is not a factor of 3915. 2.1 Factorizations a ADetermine by long division whether 12 is a factor of 3915. Slide 5Copyright 2011 Pearson Education, Inc. Solution
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EXAMPLE Solution Check sequentially the numbers 1, 2, 3, and so on, to see if we can form any factorizations. 1 72 2 36 3 24 4 18 6 12 8 9 2.1 Factorizations a Determine whether one number is a factor of another, and find the factors of a number. BFind all the factors of 72. Slide 6Copyright 2011 Pearson Education, Inc.
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Multiples A multiple of a natural number is a product of that number and some natural number. We find multiples of 2 by counting by twos: 2, 4, 6, 8, and so on. We can find multiples of 3 by counting by threes: 3, 6, 9, 12, and so on. 2.1 Factorizations b Find some multiples of a number, and determine whether a number is divisible by another. Slide 7Copyright 2011 Pearson Education, Inc.
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EXAMPLE 1 7 = 7 2 7 = 14 3 7 = 21 4 7 = 28 5 7 = 35 6 7 = 42 Solution 2.1 Factorizations b Find some multiples of a number, and determine whether a number is divisible by another. CMultiply by 1, 2, 3,… and so on, to find 6 multiples of seven. Slide 8Copyright 2011 Pearson Education, Inc.
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Thus, 15 is divisible by 5 because 15 is a multiple of 5 (15 = 3 5) 40 is divisible by 4 because 40 is a multiple of 4 (40 = 10 4) 2.1 Factorizations Divisibility Slide 9Copyright 2011 Pearson Education, Inc. The number a is divisible by another number b if there exists a number c such that a = b c. The statements “ a is divisible by b,” “ a is a multiple of b,” and “ b is a factor of a ” all have the same meaning.
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EXAMPLE Not 0 Since the remainder is not 0 we know that 102 is not divisible by 4. Solution 2.1 Factorizations b Find some multiples of a number, and determine whether a number is divisible by another. DDetermine whether 102 is divisible by 4. Slide 10Copyright 2011 Pearson Education, Inc.
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2.1 Factorizations Prime and Composite Numbers Slide 11Copyright 2011 Pearson Education, Inc. A natural number that has exactly two different factors is called a prime number. The number 1 is not prime A natural number, other than 1, that is not prime is composite.
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EXAMPLE 8 13 24 33 85 97 Determine whether the numbers listed below are prime, composite, or neither. Has factors 1, 2, 3, 4, 6, 8, 12, 24, composite Has factors of 1, 2, 4 and 8, composite Has only two factors 1 and itself, prime Has factors 1, 3, 11, 33, composite Has only two factors 1 and itself, prime 2.1 Factorizations c Given a number from 1 to 100, tell whether it is prime, composite, or neither. E Slide 12Copyright 2011 Pearson Education, Inc. Has 5 as a factor, composite
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2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157 2.1 Factorizations A Table of Primes from 2 to 157 Slide 13Copyright 2011 Pearson Education, Inc.
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EXAMPLE Solution a) Since 50 is even, it must have 2 as a factor. b) Since 25 ends in 5, we know 5 is a factor. Because 5 is prime, we can factor no further. The prime factorization can be written as 2 5 5 or 2 5 2. 2.1 Factorizations d Find the prime factorization of a composite number. FFind the prime factorization of 50. Slide 14Copyright 2011 Pearson Education, Inc.
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2.1 Factorizations d Find the prime factorization of a composite number. Slide 15Copyright 2011 Pearson Education, Inc. Every number has just one (unique) prime factorization.
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EXAMPLE · · · · = 48 Find the prime factorization of 48 using a factor tree. Had we begun with different factors (2 ∙ 24, or 4 ∙ 12), the same prime factorization would result. 2 2 2 23 6 8 2 3 4 48 2 2.1 Factorizations d Find the prime factorization of a composite number. G Slide 16Copyright 2011 Pearson Education, Inc. Solution
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EXAMPLE Solution 2.1 Factorizations d Find the prime factorization of a composite number. HFind the prime factorization of 220. Slide 17Copyright 2011 Pearson Education, Inc.
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EXAMPLE 1424 = 2 2 2 2 89 Solution We use a string of successive divisions. 2.1 Factorizations d Find the prime factorization of a composite number. IFind the prime factorization of 1424. Slide 18Copyright 2011 Pearson Education, Inc.
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