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Least Common Multiples

Least Common Multiples. Notes for August 13. Definition. Multiple – the product of two whole numbers Example: the multiples of 2 are 2, 4, 6, 8, 10, 12, …. How do you find multiples?. To find the multiples of 3, we multiply 3 by different numbers: 3 x 1 = 3 3 x 2 = 6 3 x 3 = 9

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Least Common Multiples

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  1. Least Common Multiples Notes for August 13

  2. Definition • Multiple – the product of two whole numbers • Example: the multiples of 2 are 2, 4, 6, 8, 10, 12, …

  3. How do you find multiples? • To find the multiples of 3, we multiply 3 by different numbers: • 3 x 1 = 3 • 3 x 2 = 6 • 3 x 3 = 9 • 3 x 4 = 12 • 3 x 5 = 15

  4. Find the first 4 multiples of the following numbers: • 4 • 10 • 8 • 9

  5. HELPFUL HINT #1!!! • Factors and multiples are OPPOSITES of each other • For example: 4 is a factor of 20 • Therefore, 20 is a multiple of 4

  6. Definition • Common Multiples – multiples shared by one or more numbers • For example: • 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, … • 9: 9, 18, 27, 36, 45, 54, 63, …

  7. Definition • Least Common Multiple (LCM) – the smallestmultiple that two or more numbers share in common • Example: 18 is the least common multiple of 6 and 9

  8. How do we find the LCM? • There are 2 ways to find the least common multiple: 1.) List the multiples 2.) Prime factorization – Venn Diagram • NOTE: there are other ways to find the LCM, but these are the two ways I am teaching you

  9. List the Multiples • If you want to find the LCM, you can just list all the multiples 1.) Find the LCM of 10 and 12 The multiples of 10 include: 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120, 130, 140, 150, 160, … The multiples of 12 include: 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 132, 144, … The LCM of 10 and 12 is 120

  10. Prime Factorization – Venn Diagram • To use this method, follow these steps: 1.) Find the prime factorization of each number 2.) Circle the common prime factors 3.) Square the different prime factors 4.) Write the common prime factors in the shared part of the Venn Diagram REMEMBER TO LIST EACH COMMON PRIME FACTOR ONLY ONCE! 5.) Write the different prime factors on either side of the Venn Diagram 6.) Multiply all numbers in the Venn Diagram together

  11. Prime Factorization – Venn Diagram • Find the LCM of 20 and 24 1.) Find the prime factorization of each number 24 20 2 12 2 10 6 2 2 5 3 2

  12. Prime Factorization – Venn Diagram 1.) The prime factorizations – DO NOT WRITE IN EXPONENT FORM: 20 = 2 x 2 x 5 24 = 2 x 2 x 2 x 3 2.) Circle the common prime factors 20 = 2 x 2 x 5 24 = 2 x 2 x 2 x 3

  13. Prime Factorization – Venn Diagram 3.) Square the different prime factors 20 = 2 x 2 x 5 24 = 2 x 2 x 2 x 3

  14. Prime Factorization – Venn Diagram 4.) Write all of the common prime factors in the center of the Venn Diagram and the different prime factors on either side of the Venn Diagram

  15. 24 20 2 2 5 2 3

  16. Prime Factorization – Venn Diagram 5.) Multiply the numbers in the Venn Diagram together 2 x 2 x 5 x 2 x 3 = 120 The LCM of 20 and 24 is 120

  17. HELPFUL HINT #2!!! • If you do the prime factorization of a number and there is no common prime factor, the least common multiple is the product of the two numbers. • Example: Find the LCM of 15 and 16 • The prime factorization of 15 is 3 x 5 • The prime factorization of 16 is 2 x 2 x 2 x 2 • There is no common factor. Therefore, the LCM of 15 and 16 is 15 x 16 or 240

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