172 Chapter 3 Fractions and Mixed Numbers: Addition and Subtraction 2. Finding the LCM by Using Prime Factors In Example 1 we used the method of listing multiples to find the LCM of two or more numbers.As you can see, the solution to Example 1(b) required several long lists of multiples. Here we offer another method to find the LCM of two given numbers by using their prime factors. Using Prime Factors to Find the LCM of Two Numbers Step 1 Write each number as a product of prime factors. Step 2 The LCM is the product of unique prime factors from both numbers. Use repeated factors the maximum number of times they appear in either factorization. This process is demonstrated in Example 2. Finding the LCM by Using Prime Factors Find the LCM. a. 14 and 12 b. 50 and 24 c. 45, 54, and 50 Solution: a. Find the prime factorization for 14 and 12. LCM = 2 2 3 7 = 84 b. Find the prime factorization for 50 and 24. 2’s 3’s 5’s 50 2 5 5 24 2 2 2 3 LCM = 2 2 2 3 5 5 = 600 (The LCM can also be written as 23 3 52.) c. Find the prime factorization for 45, 54, and 50. LCM = 2 3 3 3 5 5 = 1350 (The LCM can also be written as 2 33 52.) Example 2 For the factors of 2, 3, and 7, we circle the greatest number of times each occurs. The LCM is the product. The factor 5 is repeated twice. The factor 2 is repeated 3 times. The factor 3 is used only once. TIP: The product 2 2 3 7 can also be written as 22 3 7. Answers 4. 72 5. 144 6. 1260 2’s 3’s 7’s 14 2 7 12 2 2 3 2’s 3’s 5’s 45 3 3 5 54 2 3 3 3 50 2 5 5 Skill Practice Find the LCM by using prime factors. 4. 9 and 24 5. 16 and 9 6. 36, 42, and 30
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