1. The Number System Least Common Multiples 1 © 2013 Meredith S. Moody

2. Objective: You will be able to… Find multiples for a given number Find multiples for a given number Find common multiples for a pair or set of numbers Find common multiples for a pair or set of numbers Find the first (lowest) common multiple for a pair or set of numbers Find the first (lowest) common multiple for a pair or set of numbers 2 © 2013 Meredith S. Moody

3. Vocabulary Common multiple: A number exactly divisible by each number in a given set Common multiple: A number exactly divisible by each number in a given set Least common multiple: The lowest multiple exactly divisible by each number in a given set Least common multiple: The lowest multiple exactly divisible by each number in a given set 3 © 2013 Meredith S. Moody

4. Guided practice 1, multiples of single numbers Find the multiples of 10 up to 100 Find the multiples of 10 up to 100 10, 20, 30, 40, 50, 60, 70, 80, 90, & 10010, 20, 30, 40, 50, 60, 70, 80, 90, & 100 Find multiples by using multiplication: 10 x 1, 10 x 2...Find multiples by using multiplication: 10 x 1, 10 x 2... Find the multiples of 5 up to 50 Find the multiples of 5 up to 50 5, 10, 15, 20, 25, 30, 35, 40, 45, 505, 10, 15, 20, 25, 30, 35, 40, 45, 50 Find by using multiplication: 5 x 1, 5 x 2…Find by using multiplication: 5 x 1, 5 x 2… Find the multiples of 7 up to 100 Find the multiples of 7 up to 100 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 987, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 98 Find by using multiplication: 7 x 1, 7 x 2…Find by using multiplication: 7 x 1, 7 x 2… 4 © 2013 Meredith S. Moody

5. Example 1, finding least common multiples, ‘chart’ method Find the least common multiple of 2 and 3 Find the least common multiple of 2 and 3 Continue listing multiples until there is a common multiple Continue listing multiples until there is a common multiple 2 2 x 1 = 2 2 x 2 = 4 2 x 3 = 6 3 3 x 1 = 3 3 x 2 = 6 3 x 3 = 9 5 © 2013 Meredith S. Moody

6. Finding least common multiples, ‘prime factorization’ method We can use prime factorization to find the least common multiple for a set of numbers We can use prime factorization to find the least common multiple for a set of numbers Break down each number in the set into its prime factors Break down each number in the set into its prime factors For each prime factor, use the highest frequency, then find the product of the primes For each prime factor, use the highest frequency, then find the product of the primes 6 © 2013 Meredith S. Moody

7. Example 1, finding least common multiples, ‘prime factorization’ method Use prime factorization to find the least common multiple for 8 and 12 Use prime factorization to find the least common multiple for 8 and 12 The prime factorization of 8 is 2 x 2 x 2 The prime factorization of 8 is 2 x 2 x 2 The prime factorization of 12 is 2 x 2 x 3 The prime factorization of 12 is 2 x 2 x 3 The prime factor 2 appears three times in the factorization of 8 (most frequent) The prime factor 2 appears three times in the factorization of 8 (most frequent) The prime factor 3 appears one time in the factorization of 12 (most frequent) The prime factor 3 appears one time in the factorization of 12 (most frequent) Multiply 2 x 2 x 2 x 3 = 24 Multiply 2 x 2 x 2 x 3 = 24 The least common multiple of 8 and 12 is 24 The least common multiple of 8 and 12 is 24 7 © 2013 Meredith S. Moody

8. Guided practice 1, least common multiple Find the least common multiple for 6 and 8 Find the least common multiple for 6 and 8 These numbers are small, so using a chart makes the most senseThese numbers are small, so using a chart makes the most sense The least common multiple for 6 & 8 is 24The least common multiple for 6 & 8 is 24 8 6 x 1 = 66 x 2 = 126 x 3 = 186 x 4 = 24 8 x 1 = 88 x 2 = 168 x 3 = 248 x 4 = 32 © 2013 Meredith S. Moody

9. Guided practice 2, least common multiple Find the least common multiple for 12 and 30 Find the least common multiple for 12 and 30 These numbers are medium-sized; either method worksThese numbers are medium-sized; either method works OR, the prime factorization of 12 is 2 x 2 x 3; the prime factorization of 30 is 2 x 3 x 5OR, the prime factorization of 12 is 2 x 2 x 3; the prime factorization of 30 is 2 x 3 x 5 The prime factor 2 occurs once in 30 but twice in 12, so twice is the highest frequencyThe prime factor 2 occurs once in 30 but twice in 12, so twice is the highest frequency The prime factor 3 occurs once in both numbers, so once is the highest frequencyThe prime factor 3 occurs once in both numbers, so once is the highest frequency The prime factor 5 occurs once in 30, so once is the highest frequencyThe prime factor 5 occurs once in 30, so once is the highest frequency 2 x 2 x 3 x 5 = 602 x 2 x 3 x 5 = 60 9 12 x 1 = 1212 x 2 = 2412 x 3 = 3612 x 4 = 4812 x 5 = 60 30 x 1 = 3030 x 2 = 6030 x 3 = 9030 x 4 = 12030 x 5 = 150 © 2013 Meredith S. Moody

10. Guided practice 3, least common multiple Find the least common multiple for 25 and 75 Find the least common multiple for 25 and 75 These numbers are large, so using the prime factorization method makes the most senseThese numbers are large, so using the prime factorization method makes the most sense The prime factorization of 25 is 5 x 5The prime factorization of 25 is 5 x 5 The prime factorization of 75 is 3 x 5 x 5The prime factorization of 75 is 3 x 5 x 5 The prime factor 3 occurs once in 75; once is the highest frequencyThe prime factor 3 occurs once in 75; once is the highest frequency The prime factor 5 occurs twice in both 25 and 75; twice is the highest frequencyThe prime factor 5 occurs twice in both 25 and 75; twice is the highest frequency 3 x 5 x 5 = 753 x 5 x 5 = 75 75 is the least common multiple for 25 and 7575 is the least common multiple for 25 and 75 10 © 2013 Meredith S. Moody

11. Guided practice 4, least common multiple A local restaurant is offering a free meal to every 25 th customer and a free hat to every 12 th customer. Which customer will be the first to get both a free meal and a free hat? A local restaurant is offering a free meal to every 25 th customer and a free hat to every 12 th customer. Which customer will be the first to get both a free meal and a free hat? Find the least common multiple Find the least common multiple Prime factorization of 25 is 5 x 5 Prime factorization of 25 is 5 x 5 Prime factorization of 12 is 2 x 2 x 3 Prime factorization of 12 is 2 x 2 x 3 2 x 2 x 3 x 5 x 5 = 300 2 x 2 x 3 x 5 x 5 = 300 The 300 th customer will get both items The 300 th customer will get both items 11 © 2013 Meredith S. Moody

12. Common mistakes One mistake people make when finding the least common multiple is to simply multiply the two numbers together One mistake people make when finding the least common multiple is to simply multiply the two numbers together This DOES give A common multiple, but it might not always be the LEAST common multiple This DOES give A common multiple, but it might not always be the LEAST common multiple People usually do this because they are lazy!...or don’t know any better People usually do this because they are lazy!...or don’t know any better 12 © 2013 Meredith S. Moody

13. Common mistakes Another common mistake is that people find the greatest common factor INSTEAD of the least common multiple and vice versa Another common mistake is that people find the greatest common factor INSTEAD of the least common multiple and vice versa How do you remember what to do? How do you remember what to do? One way to remember is that greatest common factor has “factor” in it – this sounds like “fraction” and fractions are smaller than whole numbers  factors are smaller than the original numbers (at least one of them) One way to remember is that greatest common factor has “factor” in it – this sounds like “fraction” and fractions are smaller than whole numbers  factors are smaller than the original numbers (at least one of them) Least common multiple has the word “multiple” in it – this sounds like “multiply” and when we multiply we get larger numbers  common multiples are larger than the original numbers (at least one of them) Least common multiple has the word “multiple” in it – this sounds like “multiply” and when we multiply we get larger numbers  common multiples are larger than the original numbers (at least one of them) © 2013 Meredith S. Moody 13

14. Least common multiple: You try Find the least common multiple of the following sets of numbers Find the least common multiple of the following sets of numbers 4 & 10 4 & 10 2020 12, 15, & 20 12, 15, & 20 6060 4 & 18 4 & 18 3636 14 © 2013 Meredith S. Moody