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

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.

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.

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 Factorizations a ADetermine by long division whether 12 is a factor of Slide 5Copyright 2011 Pearson Education, Inc. Solution

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  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.

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.

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.

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.

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.

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.

EXAMPLE 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

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, Factorizations A Table of Primes from 2 to 157 Slide 13Copyright 2011 Pearson Education, Inc.

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  Factorizations d Find the prime factorization of a composite number. FFind the prime factorization of 50. Slide 14Copyright 2011 Pearson Education, Inc.

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.

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 Factorizations d Find the prime factorization of a composite number. G Slide 16Copyright 2011 Pearson Education, Inc. Solution

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.

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 Slide 18Copyright 2011 Pearson Education, Inc.