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Math 35 Fall Term Professor Carl Scarbnick scarbnick@gmail.com Topic: Least Common Multiples

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Section 1.6 Least Common Multiples Objectives 1 1. Find the least common multiple (LCM) 2. Find the LCM using multiples of the largest number 3. Find the LCM using prime factorization (Warning: The graphics in these slides depend on the screen resolution of your monitor. Please use the handout passed out on Aug 27, 2007 if you experience any problems.)

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2 1.6 Least Common Multiples Objective 1. Find the least common multiples The least common multiple (LCM) of two whole numbers is the smallest number divisible by both these numbers.

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3 1.6 Least Common Multiples Objective 1. Find the least common multiples Example: Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36 Multiples of 7: 7, 14, 21, 28, 35, 42, 49, 56, 63 The smallest number in each list is 28. This means that 28 is the least common multiple (LCM) of 4 and 7.

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4 1.6 Least Common Multiples Objective 2. Find the LCM using multiples of the largest number When the numbers are small, we can find the LCM with the following procedure Write out the multiples of the larger number and find the smallest one that is a multiple of the smaller number.

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5 1.6 Least Common Multiples Objective 2. Find the LCM using multiples of the largest number Example: Find the least common multiple of 5 and 6. The multiples of 6 are 6, 12, 18, 24, 30, 36, … The smallest multiple where the last digit is zero or five is the number, 30. Therefore, 30 is the least common multiple of 5 and 6.

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6 1.6 Least Common Multiples Objective 3. Find the LCM using prime factorization. Example: The prime factorization of 6 = 2 x 3. The prime factorization of 9 = 3 x 3. Notice that the prime factorization of 6 and the prime factorization of 9 includes the number, 3.

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7 1.6 Least Common Multiples Objective 3. Find the LCM using prime factorization. Example: The prime factorization of 6 = 2 x 3. The prime factorization of 9 = 3 x 3. The LCM of 6 and 9 is 2 x 3 x 3 = 18. Factors of 6 Factors of 9

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8 1.6 Least Common Multiples Objective 3. Find the LCM using prime factorization. Example: The LCM of 6 and 9 is 2 x 3 x 3 = 18. We do not repeat the prime factors shared by the numbers, 6 and 9. Note that 6 x 9 = (2 x 3) x (3 x 3) = 54 is a multiple of 6 and 9. However, it is not the least common multiple, since it repeats a prime factor shared by both numbers.

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9 1.6 Least Common Multiples Objective 3. Find the LCM using prime factorization. General Procedure for computing the LCM with prime factorization: Step 1: Write the prime factorization of each number. Step 2: Take the product of the prime numbers that appear in each list. If a prime number appears more than one time in a list, use it the most number of times it appears in a list.

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10 1.6 Least Common Multiples Objective 3. Find the LCM using prime factorization. Problem 1: Find the least common multiple of 14 and 36 Step 1: The prime factorization of 14 is 14 = 2 x 7 The prime factorization of 36 is 36 = 2 x 2 x 3 x 3 Step 2 LCM = 2 x 2x 3 x 3 x 7 = 252 Prime Numbers237 Most number of times in a list 221

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11 1.6 Least Common Multiples Objective 3. Find the LCM using prime factorization. Problem 2: Find the least common multiple of 24 and 62. Step 1: The prime factorization of 24 is 24 = 2 x 2 x 2 x 3 The prime factorization of 62 is 62 = 2 x 31 Step 2 LCM = 2 x 2 x 2 x 3 x 31 = 744 Prime Numbers2331 Most number of times in a list 311

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