5-1 Least Common Multiple Learn to find the least common multiple (LCM) of a group of numbers.

5-1 Least Common Multiple Vocabulary least common multiple (LCM) ‏

5-1 Least Common Multiple A multiple of a number is the product of the number and any nonzero whole number. Remember!

5-1 Least Common Multiple Additional Example 1: Consumer Application English muffins come in packs of 8, and eggs come in cartons of 12. If there are 24 students, what is the least number of packs and cartons needed so that each student has a muffin sandwich with one egg and there are none left over? There are 24 English muffins and 24 eggs. So 3 packs of English muffins and 2 cartons of eggs are needed. Draw muffins in groups of 8. Draw eggs in groups of 12. Stop when you have drawn the same number of each.

5-1 Least Common Multiple Check It Out: Example 1 There are 18 dog cookies and 18 bones. So 3 packages of dog cookies and 2 bags of bones are needed. Dog cookies come in packages of 6, and bones in bags of 9. If there are 18 dogs, what is the least number of packages and bags needed so that each dog has a treat box with one bone and one cookie and there are no bones or cookies left over? Draw cookies in groups of 6. Draw bones in groups of 9. Stop when you have drawn the same number of each.

5-1 Least Common Multiple The smallest number that is a multiple of two or more numbers is the least common multiple (LCM). In Additional Example 1, the LCM of 8 and 12 is 24.

5-1 Least Common Multiple Additional Example 2A: Using Multiples to Find the LCM Find the least common multiple (LCM). Method 1: Use a number line. 3 and 4 Use a number line to skip count by 3 and The least common multiple (LCM) of 3 and 4 is 12.

5-1 Least Common Multiple Additional Example 2B: Using Multiples to Find the LCM Find the least common multiple (LCM). Method 2: Use a list. 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44,... 5: 5, 10, 15, 20, 25, 30, 35, 40, 45,... 8: 8, 16, 24, 32, 40, 48,... LCM: 40 List multiples of 4, 5, and 8. Find the smallest number that is in all the lists. 4, 5, and 8

5-1 Least Common Multiple The prime factorization of a number is the number written as a product of its prime factors. Remember!

5-1 Least Common Multiple Additional Example 2C: Using Multiples to Find the LCM Find the least common multiple (LCM). Method 3: Use prime factorization. 6 and 20 6 = = Write the prime factorization of each number. Line up the common factors = 60 To find the LCM, multiply one number from each column. LCM: 60

5-1 Least Common Multiple Additional Example 2D: Using Multiples to Find the LCM Find the least common multiple (LCM). 15, 6, and 4 15 = = 2 2 Write the prime factorization of each number in exponential form = 60 To find the LCM, multiply each prime factor once with the greatest exponent used in any of the prime factorizations. 6 = 3 2 LCM: 60

5-1 Least Common Multiple Check It Out: Example 2A Find the least common multiple (LCM). Method 1: Use a number line. 2 and 3 Use a number line to skip count by 2 and The least common multiple (LCM) of 2 and 3 is 6.

5-1 Least Common Multiple Check It Out: Example 2B Find the least common multiple (LCM). Method 2: Use a list. 3, 4, and 9 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36,... 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, … 9: 9, 18, 27, 36, 45,... The least common multiple of 3, 4, and 9 is 36. List multiples of 3, 4, and 9. Find the smallest number that is in all the lists.

5-1 Least Common Multiple Check It Out: Example 2C Find the least common multiple (LCM). Method 3: Use prime factorization. 4 and 10 4 = = 2 5 Write the prime factorization of each number. Line up the common factors = 20 To find the LCM, multiply one number from each column. LCM: 20

5-1 Least Common Multiple Check It Out: Example 2D Find the least common multiple (LCM). 12, 6, and 8 12 = = 2 3 Write the prime factorization of each number in exponential form = 24 To find the LCM, multiply each prime factor once with the greatest exponent used in any of the prime factorizations. 8 = 2 3 LCM: 24