Copyright © Cengage Learning. All rights reserved. 1.5 Prime Factorization

22 Divisibility

33 A number is divisible by a second number if, when you divide the first number by the second number, you get a zero remainder. That is, the second number divides the first number. Example 1 12 is divisible by 3, because 3 divides 12. Divisibility

44 An integer is even if it is divisible by 2; that is, if you divide it by 2, you get a zero remainder. An integer is odd if it is not divisible by 2. In multiplying two or more positive integers, the positive integers are called the factors of the product. Thus, 2 and 5 are factors of 10, since 2  5 = 10. The numbers 2, 3, and 5 are factors of 30, since 2  3  5 = 30. Divisibility

55 Divisibility Tests

66 The following divisibility tests for certain positive integers are most helpful. Divisibility by 2 If a number ends with an even digit, then the number is divisible by 2. Note Zero is even. Divisibility Tests

77 Is 4258 divisible by 2? Yes; since 8, the last digit of the number, is even, 4258 is divisible by 2. Example 7

88 Divisibility by 3 If the sum of the digits of a number is divisible by 3, then the number itself is divisible by 3. Example 9 Is 531 divisible by 3? The sum of the digits = 9. Since 9 is divisible by 3, then 531 is divisible by 3. Divisibility Tests

99 Divisibility by 5 If a number has 0 or 5 as its last digit, then the number is divisible by 5. Example 11 Is 2372 divisible by 5? The last digit of 2372 is neither 0 nor 5, so 2372 is not divisible by 5. Divisibility Tests

10 There are many ways of classifying the positive integers. They can be classified as even or odd, as divisible by 3 or not divisible by 3, as larger than 10 or smaller than 10, and so on. One of the most important classifications involves the concept of a prime number: an integer greater than 1 that has no divisors except itself and 1. The first ten prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29. Divisibility

11 Any number that is not prime can be broken down into a series of prime factors. The process of finding the prime factors of a positive integer is called prime factorization. The prime factorization of a given number is the product of its prime factors. That is, each of the factors is prime, and their product equals the given number. One of the most useful applications of prime factorization is in finding the least common denominator (LCD) when adding and subtracting fractions. Divisibility

12 1.Find a prime number that divides the given number. Divide, using short division. 2.Then find a second prime number that divides the result. Divide, keeping both factors listed. 3. Keep repeating this process of listing the quotients and divisors until the final quotient is also prime. 4.The prime factors will be the product of the divisors and the final quotient of the repeated short divisions. How to find the Prime Factorization

13 To eliminate some of the guesswork involved in finding prime factors, divisibility tests can be used. Divisibility tests and prime factorization are used to reduce fractions to lowest terms and to find the lowest common denominator. Divisibility Tests

14 Find the prime factorization of 45. The prime factorization of 45 is 3  3  5. You may also use factor trees: Example 4 Divide by 3. (2,3, and 5 are the best to keep using) Divide by 3.(always start with a prime number!)

15 Note As a general rule of thumb: 1. Keep dividing by 2 until the quotient is not even. OR 2. Keep dividing by 3 until the quotient’s sum of digits is not divisible by 3. OR 3. Keep dividing by 5 until the quotient does not end in 0 or 5. That is, if you divide out all the factors of 2, 3, and 5, the remaining factors, if any, will be much smaller and easier to work with, and perhaps prime. Divisibility Tests

16 Recap Video (Khan Academy) ic/factors- multiples/prime_factorization/v/prime- factorization

17 Problems for Discussion 1.5 Pg odds 45 hint: divide by 5 first 52 check divisibility tests and remember what a prime number is 59 start with dividing by 3 Why? 64 start with dividing by 2 Why?