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

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5-1 Least Common Multiple

Least Common Multiple

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A multiple of a number is the product of the number and any nonzero whole number.

Remember!

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When thinking about finding the When thinking about finding the Least Common Multiple, or the Least Common Multiple, or the

LCM…LCM…

REMEMBERREMEMBER

LL…The …The smallestsmallest number number

CC…all terms in …all terms in commoncommon

MM…from their list of …from their list of multiplesmultiples

Least Common Multiple Least Common Multiple (LCM)(LCM)

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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 no muffins 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.

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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.

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The smallest number that is a multiple of two or more numbers is the least common multiple (LCM).

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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 4.

0 2 4 6 8 10 12

The least common multiple (LCM) of 3 and 4 is 12.

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Additional Example 2B: Using Multiples to Find the 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

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Check It Out: Example 2A


Method 1: Use a number line.

2 and 3Use a number line to skip count by 2 and 3.

0 1 2 3 4 5 6

The least common multiple (LCM) of 2 and 3 is 6.

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Check It Out: Example 2B


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.

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Finding the LCM: Method 1 – Multiple ListFinding the LCM: Method 1 – Multiple ListExample 1Example 1:: Find the LCM of 4 and 9. Find the LCM of 4 and 9.

Step 1: Create a list of multiples for each numberStep 2: Circle the first multiple the numbers have in common

44: 4, 8, 12, 16, 20, 24, 28, 32, : 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, …36, 40, …

The LCM of 4 and 9 is 36


99: 9, 18, 27, 36, 45, 54, 63,…: 9, 18, 27, 36, 45, 54, 63,…

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Finding the LCM: Method 2 – Prime Finding the LCM: Method 2 – Prime FactorizationFactorization

Example 1Example 1:: Find the LCM of 4 and 9. Find the LCM of 4 and 9.Step 1: Find the prime factorization of each number.Step 2: Find the product of all the prime factors.

44 99

LCM = 36

22

2 · 2 · 22

2244: : 22 · · 2299: : 33 · · 33

22 ·· 2 ·2 · 33 ·· 3 3 = 36= 36


33

3 · 3 · 33

33

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Finding the LCM: Method 3 – Find the ProductFinding the LCM: Method 3 – Find the Product

Example 1Example 1:: Find the LCM of 4 and 9. Find the LCM of 4 and 9.Step 1: For any two numbers, multiply them togetherStep 2: Your answer is the LCM of the numbers


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Step 1: Create a list of multiple for each numberStep 2: Circle the first multiple the numbers have in common

1010: 10, 20, 30, 40, 50, 60, 70, : 10, 20, 30, 40, 50, 60, 70, 80,…80,…

The LCM of 10 and 12 is 60

1212: 12, 24, 36, 48, 60, 72, 84,: 12, 24, 36, 48, 60, 72, 84,……

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Step 1: Create a list of multiple for each numberStep 2: Circle the first multiple the numbers have in common

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Guided Practice ProblemsGuided Practice ProblemsDirectionsDirections:: Find the LCM for each Find the LCM for each

number set.number set.

1. 9 and 121. 9 and 12

2. 4, 8, and 122. 4, 8, and 12

3. 2, 3, 6, and 83. 2, 3, 6, and 8


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Lesson Quiz


1. 6, 14 2. 9, 12

3. 5, 6, 10 4. 12, 16, 24, 36

42 36

30 144

Documents

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