Copyright © Ed2Net Learning, Inc.1 Least Common Multiple Grade 6

2 Find the GCF of 1) 160 and 550 2) 36 and 48 Find the GCF of each set of numbers by the listing the common prime factors of each number 3) 80, 110 4) 42, 49 Warm up

3 Lets review what we have learned in the last Lesson Greatest Common Factor (GCF) of two or more numbers can be defined as the greatest number that is a factor of each number Two methods can be used to find GCF

4 List the factors of each number. Then identify the common factors. The greatest of these common factors is the GCF. Write the prime factorization of each number. Then identify all common prime factors and find their product, the GCF Method 2 Method 1

5 Find the GCF of 105 and 63 by the two methods Factors of 105 : 1, 3, 5, 7, 15, 21, 35, 105 Factors of 63 : 1, 3, 7, 9, 21,63 Since the common factors are 1, 3, 7 and 21 The GCF is 21 Method 1

6 Method 2 Write the prime factorizations of each x 7 3 x 5 x x 7 3 x 3 x 7

7 In the Real World Ferry Boats Two ferry boats leave a loading platform at the same time. One of the ferry boats returns to the loading platform every 25 minutes. The other returns every 30 minutes. In the next 300 minutes, when will they return at the same time? Least Common Multiple

8 In the Real World Ferry Boats Two ferry boats leave a loading platform at the same time. One of the ferry boats returns to the loading platform every 25 minutes. The other returns every 30 minutes. In the next 300 minutes, when will they return at the same time? You can use multiples to answer the question above. A multiple of a number is the product of the number and any nonzero whole number. Multiples of 2 : 2,4,6,8,10,12,14,… Least Common Multiple The three dots show that the pattern continues forever.

9 In the Real World Ferry Boats Two ferry boats leave a loading platform at the same time. One of the ferry boats returns to the loading platform every 25 minutes. The other returns every 30 minutes. In the next 300 minutes, when will they return at the same time? You can use multiples to answer the question above. A multiple of a number is the product of the number and any nonzero whole number. Multiples of 2 : 2,4,6,8,10,12,14,… Least Common Multiple The three dots show that the pattern continues forever. A multiple shared by two or more numbers is a common multiple.

10 25, 50, 75, 100, 125, 150, 175, 200, 225, 250, 275, 300, … 30, 60, 90, 120, 150, 180, 210, 240, 270, 300, … Finding a Common Multiple ANSWER EXAMPLE 1 You can use common multiples to answer this question. Begin by writing the multiples of 25 and 30. Multiples of 25 : Multiples of 30 : Now identify the common multiples through 300. The ferry boats will return to the loading platform at the same time in 150 minutes and in 300 minutes Least Common Multiple Ferry Boats Two ferry boats leave a loading platform at the same time. One of the ferry boats returns to the loading platform every 25 minutes. The other returns every 30 minutes. In the next 300 minutes, when will they return at the same time?

11 NOTE BOOK Finding the Least Common Multiple (LCM) Method 1: Method 2: Start listing the multiples of each number. Then find the smallest of the common multiples. Write the prime factorizations of the numbers. Multiply together the prime factors, using each prime factor the greatest number of times it is a factor of any of the numbers. The least common multiple of two or more numbers is the smallest of the common multiples. Below are two methods to find the LCM. Least Common Multiple

12 9, 18, 27, 36, 45, 54, … 12, 24, 36, 48, … Find the LCM of 9 and 12. Finding the LCM EXAMPLE 2 Multiples of 9 : Multiples of 12 : 36 ANSWER The LCM of 9 and 12 is 36. Least Common Multiple

13 Find the LCM of 42 and 60 using prime factorization. Using Prime Factorization EXAMPLE 3 Write the prime factorizations. Circle any common factors. Multiply together the prime factors, using each circled factor the greatest number of times it occurs in either factorization. 42 = 2  3  7 60 = 2  2  3  5 2  2  3  5  7 ANSWER The LCM of 42 and 60 is 420. = Least Common Multiple

14 Lets take an Example Use method 2 to find the LCM of 9, 12 and 15 9 = 3 x 3 = = 2 x 2 x 3 = 2 2 x 3 15 = 3 x 5

15 The prime factors are 2, 3 and 5. The greatest number of times 2 appears is twice (in 12), So we write it down twice. The greatest number of times 3 appears is twice (in 9), so we again write it down twice. The greatest number of times 5 appears is once (in 15), so write it down once. The LCM of 9, 12 and 15 is 2 x 2 x 3 x 3 x 5 = 180 Example Conti…

16 Find the LCM of 8 and 12 Multiples of 8 : 8, 16, 24, 32, 40,… Multiples of 12 : 12, 24, 36, 48, 60,… The LCM of 8 and 12 is 24 8 = 2 x 2 x 2 12 = 2 x 2 x 3 The LCM of 8 and 12 is 2 x 2 x 2 x3 = 24 Write the prime factors of each METHOD 1METHOD 2

17 Find the LCM of each set of numbers using multiples 1)9, 10, 4 2) 30, 15 3) Mentally compute the LCM of 4, 5 and 10 4) Use calculator to find the LCM of 56 and 16 Try These!

18 Let us take a break!!

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20 Serena plans a fitness program that will include bicycling every 3 days, hiking every 7 days, and swimming every 14 days. How many days will pass before she will do all 3 activities on the same day?

21 Write a set of three numbers whose LCM is the product of the numbers

22 Margo has piano lessons every two weeks. Her brother Roberto has a soccer tournament every 3 weeks. Her sister Rocio has an orthodontist appointment every four weeks. If they all have this activity on Friday, How long will it be before their activities fall on the same day again? Critical Thinking

23 We use two methods to compute the LCM The least common multiple of the numbers a and b is the smallest number that is divisible by both a and b. We denote the least common multiple of a and b by lcm (a, b). Review

24 List several multiples of each number. Then identify the common multiples. The least of these is the LCM. Method 1 Find the prime factors of each number, then identify all common prime factors. For each prime factor, write it down the greatest number of times it appears in any of the numbers. The product is the LCM Method 2

25 Lets see some Examples Compute the LCM for each set of numbers 1) 29, 58, ) 48, 16, 3 48

26 Great Job Done Be sure to practice what you have learned today