110 Chapter 2 Fractions and Mixed Numbers: Multiplication and Division Determining Prime Factorizations Example 5 Find the prime factorization. a. 198 b. 153 Solution: a. The sum of the digits 9 9 18 is divisible by 3. 2 3 3 11 2 32 11. Since 198 is even, we know it is divisible by 2. 2)198 3)99 3)33 11 The prime factorization of 198 is or 3)153 3)51 17 5. Identifying All Factors of a Whole Number Sometimes it is necessary to identify all factors (both prime and other) of a number. Take the number 30, for example. A list of all factors of 30 is a list of all whole numbers that divide evenly into 30. Factors of 30: 1, 2, 3, 5, 6, 10, 15, and 30 b. The prime factorization of 153 is 3 3 17 or 32 17. Listing All Factors of a Number Example 6 List all factors of 36. Solution: Begin by listing all the two-number factorizations of 36. This can be accomplished by systematically dividing 36 by 1, 2, 3, and so on. Notice, however, that after the product 6 6 , the two-number factorizations are repetitious, and we can stop the process. These products repeat the factorizations above.Therefore, we can stop at 1 36 2 18 3 12 4 9 6 6 9 4 12 3 ⎫⎪⎬⎪⎭ 18 2 6 6. 36 1 The list of all factors of 36 consists of the individual factors in the products. The factors are 1, 2, 3, 4, 6, 9, 12, 18, 36. Answers 10. 11. 12. 1, 3, 5, 9, 15, 45 2 2 2 3 7 or 23 3 7 2 3 3 5 11 or 2 32 5 11 TIP: When listing a set of factors, it is not necessary to write the numbers in any specified order. However, in general we list the factors in order from smallest to largest. Skill Practice Find the prime factorization of the given number. 10. 168 11. 990 Skill Practice 12. List all factors of 45.
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