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Factoring Integers November 3, 2011 Factoring Integers Objective To factor integers and to find the greatest common factor of several integers. Factoring Integers When you write 56 = 8 ∙ 7 or 56 = 4 ∙ 14, you have factored 56. In the first case the factors are 8 and 7. In the second case the factors are 4 and 14. You could also write 1 56 = ∙ 112 and call ½ and 112 factors of 2 56. Usually, however, you are interested only in factors that are integers. Factoring Integers To factor a number over a given set. you write it as a product of numbers in that set, called the factor set. In this class, integers will be factored over the set of integers unless some other set is specified. The factors are then integral factors. Positive Factors You can find the positive factors of a given positive integer by dividing it by positive integers in order. Record only the integral factors. Continue until a pair of factors is repeated. Example 1 Give all the positive factors of 56. Solution Divide 56 by l, 2, 3, 56 = 156 = 228 = 414 = 78 (= 87) Stop the positive factors of 56 are 1, 2, 4, 7, 8, 14, 28, and 56. Prime Numbers A prime number, or prime, is an integer greater than 1 that has no positive integral factor other than itself and 1. Is 1 a prime number? No! 1≯1 The first ten prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 Prime Factorization To find the prime factorization of a positive integer, you express it as a product of primes. Example 2 shows a way to organize your work. Example 2 Find the prime factorization of 504. Solution 504 = 2 ∙ 252 Try the primes in = 2 ∙ 2 ∙ 126 order as divisors. = 2 ∙ 2 ∙ 2 ∙ 63 Divide by each prime = 2 ∙ 2 ∙ 2 ∙ 3 ∙ 21 as many times as possible before going =2∙2∙2∙3∙3∙7 on to the next prime. = 23 ∙ 32 ∙ 7 Exponents Exponents are generally used for prime factors that occur more than once in a factorization. The prime factorization of an integer is unique (there is only one) except for the order of the factors. Greatest Common Factor A factor of two or more integers is called a common factor of the integers. The greatest common factor (GCF) of two or more integers is the greatest integer that is a factor of all the given integers. 2 is a common factor of 6 and 16. 6 is the greatest common factor of 12 and 18. Example 3 Find the GCF of 882 and 945. Solution First find the prime factorization of each integer. Then form the product of the smaller powers of each common prime factor. Example 3 Solution 882 = 2 ∙ 32 ∙ 72 945 = 33 ∙ 5 ∙ 7 The common prime factors are 3 and 7. The smaller powers of 3 and 7 are 32 and 7. the GCF of 882 and 945 is 327, or 63. Class work p 186: Oral Exercises 1-22 Homework p 186: 3-39 mult of 3, 41-46, p 187: Mixed Review