1

Least Common MultiplesAnd Greatest Common

Factor

Least Common MultiplesAnd Greatest Common

Factor

2

Lesson

3.4.5Least Common MultiplesLeast Common Multiples

What it means for you:You’ll learn how to find the lowest number that two numbers both divide into exactly. You’ll also see how this can help you with adding and subtracting fractions.

Key words:• least common multiple• greatest common factor

California Standards:Number Sense 2.1Solve problems involving addition, subtraction, multiplication, and division of positive fractions and explain why a particular operation was used for a given situation.Number Sense 2.4Determine the least common multiple and the greatest common divisor of whole numbers; use them to solve problems with fractions (e.g., to find a common denominator to add two fractions or to find the reduced form for a fraction).

3

Least Common MultiplesLeast Common Multiples

The Least Common ulte Is Less Than the Others

Lesson

3.4.5

Today we are going to learn about Lowest Common Multiples and Greatest Common Factor. First let’s examine the parts of a multiplication equation.

MULTIPLE

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Least Common MultiplesLeast Common MultiplesLesson

3.4.5

Example: What is the LCM of 10 and 30?

10 x 1 = 1010 x 2 = 2010 x 3 = 3010 x 4 = 4010 x 5 = 5010 x 6 = 60

30 x 1 = 3030 x 2 = 6030 x 3 = 90

So the least common multiple of 10 and 30 is 30.

5

Least Common MultiplesLeast Common Multiples

The Least Common ulte Is Less Than the Others

Lesson

3.4.5

When do you use LCM?

302

101+

When you need to add fractions with different denominators!

6

Least Common MultiplesLeast Common MultiplesLesson

3.4.5

Example: What is the GCF of 20 and 30?

1 x 20 2 x 104 x 5

1 x 302 x 153 x 105 x 6

7

Least Common MultiplesLeast Common Multiples

The Least Common ulte Is Less Than the Others

Lesson

3.4.5

When do you use GCF?

3020

When you need to SIMPLIFY fractions!

8

Least Common MultiplesLeast Common Multiples

The LeastThan the Others

Lesson

3.4.5

You can find the Lowest Common Multiple (LCM) and the Greatest Common Factor (GCF)with a factor tree. Let’s try some!

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Least Common MultiplesLeast Common Multiples

Example 1

Lesson

3.4.5

Solution

Solution follows…

Find the LCM and GCF of 20 and 12.

So the least common multiple of 20 and 12 is 60 .

10

Least Common MultiplesLeast Common Multiples

Example 2

Lesson

3.4.5

Solution

Solution follows…

Find the LCM and GCF of 16 and 28.

So the least common multiple of 16 and 28 is 112.

11

Least Common MultiplesLeast Common Multiples

Example 3

Lesson

3.4.5

Solution

Solution follows…

Find the LCM and GCF of 12 and 44.

.

So the least common multiple of 12 and 44 is: 12 × 44 ÷ 4 = 132

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Least Common MultiplesLeast Common Multiples

Guided Practice

Solution follows…

Lesson

3.4.5

Find the least common multiple and greatest common factor.

1. 4, 40 2. 15, 50

3. 12, 9 4. 10, 25

13

Least Common MultiplesLeast Common Multiples

Independent Practice

Solution follows…

Lesson

3.4.5

Find the LCM and GCF.

7. 30, 45 8. 4, 12

9. 14, 21 10. 28, 30

14

LCM and GCF: Fractions with Different Denominators

LCM and GCF: Fractions with Different Denominators

15

LCM and GCFLCM and GCF

Example 1

Lesson

3.4.5

Solution:

=+ 51

31

+ 153

155

=5 + 3

15 = 158

What is + ?51

31

16

LCM and GCFLCM and GCF

Example 2

Lesson

3.4.5

What is - ?165

403

17

LCM and GCFLCM and GCF

Example 3

Lesson

3.4.5

What is 2 ½ + 3 ¼ =

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Least Common MultiplesLeast Common Multiples

Guided Practice

Solution follows…

Lesson

3.4.5

Use factor trees to find the GCF and LCM. Solve the equations once they have the same denominator.

1. 2.

3. 4.

49 +

5610

13159

+

1051

6+

56

–18

3930

1830

5730

1910

+= ==8

181518

2318

+= =

530

1530

2030

23

+= ==2024

324

1724

= =–

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Least Common MultiplesLeast Common Multiples

Guided Practice

Solution follows…

Lesson

3.4.5

Solve the following problems.

5.

6.

202

+153

4213

145

+

46

–28

1260

660

1860

310

+= ==1342

1542

2842

23

+= ==

1624

624

1024

512

– == =**TICKET

PROBLEM

20

Least Common MultiplesLeast Common Multiples

Independent Practice

Solution follows…

Lesson

3.4.5

7. 8. 9.

10. 11. 12.

13. 14. 15.

23

–14

34

512

149–10

9

14

34

–12

19

+27

89

–56

1812+15

13+11

84418

–1213

+102

2515

2563

13

45

118

2315

2522

935

Find the LCM and GCF for problems 7-15 by making factor trees. Then add or subtract the fractions.

21

Least Common MultiplesLeast Common Multiples

Independent Practice

Solution follows…

Lesson

3.4.5

Work out the sums and differences in Exercises 26–31. Give your answers in their simplest form.

26. 27. 28.

29. 30. 31.

451–27

4

5610–42

31

+757

3515

+241

5615

4910

216

+ 3017

9+

1342

274525

17135

4784

7390

2449

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Least Common MultiplesLeast Common MultiplesLesson

3.4.5

Round UpRound Up

Pausing for a second to find the least common multiple before trying to add or subtract two fractions with different denominators is a good way to avoid having to deal with really big numbers.

Smaller numbers also make simplifying at the end easier.