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Least Common Multiples And Greatest Common Factor

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Least Common Multiples

And Greatest Common Factor

Lesson

3.4.5

Least 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.

California Standards:

Number Sense 2.1

Solve problems involving addition, subtraction,multiplication, and division of positive fractions and explain why a particular operation was used for a given situation.

Number Sense 2.4

Determine 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).

- Key words:
- least common multiple
- greatest common factor

Lesson

3.4.5

Least Common Multiples

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

The Least Common ulte Is Less Than the Others

MULTIPLE

Lesson

3.4.5

Least Common Multiples

Definition

Least Common Multiple (LCM): The lowest multiple of two or more numbers.

Example: What is the LCM of 10 and 30?

10 x 1 = 10

10 x 2 = 20

10 x 3 = 30

10 x 4 = 40

10 x 5 = 50

10 x 6 = 60

30 x 1 = 30

30 x 2 = 60

30 x 3 = 90

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

2

1

30

10

Lesson

3.4.5

Least Common Multiples

When do you use LCM?

The Least Common ulte Is Less Than the Others

+

When you need to add fractions with different denominators!

Lesson

3.4.5

Least Common Multiples

Definition

Greatest Common Factor (GCF): The largest factor shared by two or more numbers.

1 x 30

2 x 15

3 x 10

5 x 6

Example: What is the GCF of 20 and 30?

1 x 20

2 x 10

4 x 5

The factors of 20 are 1, 2, 4, 5, 10, and 20

The factors of 30 are 1, 2, 3, 5, 6, 10, 15 and 30

20

30

Lesson

3.4.5

Least Common Multiples

When do you use GCF?

The Least Common ulte Is Less Than the Others

When you need to SIMPLIFY fractions!

Lesson

3.4.5

Least Common Multiples

You can find the Lowest Common Multiple (LCM) and the Greatest Common Factor (GCF)

with a factor tree. Let’s try some!

The LeastThan the Others

Lesson

3.4.5

Least Common Multiples

Example 1

Find the LCM and GCF of 20 and 12.

Solution

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

Solution follows…

Lesson

3.4.5

Least Common Multiples

Example 2

Find the LCM and GCF of 16 and 28.

Solution

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

Solution follows…

Lesson

3.4.5

Least Common Multiples

Example 3

Find the LCM and GCF of 12 and 44.

.

Solution

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

Solution follows…

Lesson

3.4.5

Least Common Multiples

Guided Practice

Find the least common multiple and greatest common factor.

1. 4, 402. 15, 50

3. 12, 94. 10, 25

Solution follows…

Lesson

3.4.5

Least Common Multiples

Independent Practice

Find the LCM and GCF.

7. 30, 458. 4, 12

9. 14, 2110. 28, 30

Solution follows…

LCM and GCF: Fractions with Different Denominators

1

1

3

5

5

1

1

3

5 + 3

8

+

=

=

=

+

5

3

15

15

15

15

Lesson

3.4.5

LCM and GCF

Example 1

What is + ?

Solution:

3

5

40

16

Lesson

3.4.5

LCM and GCF

Example 2

What is - ?

Lesson

3.4.5

LCM and GCF

Example 3

What is 2 ½ + 3 ¼ =

20

23

17

3

20

8

15

19

57

18

39

15

5

2

30

30

24

24

24

10

30

18

30

30

18

30

18

3

=

+

=

=

=

+

=

=

=

=

+

13

9

4

5

+

+

=

10

–

=

15

9

6

1

5

5

1

+

–

6

10

6

8

Lesson

3.4.5

Least Common Multiples

Guided Practice

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

1. 2.

3. 4.

Solution follows…

18

5

12

3

13

6

28

16

6

10

15

2

60

24

24

42

12

42

24

60

42

10

60

3

3

2

13

5

+

+

=

+

=

=

=

+

=

=

15

20

42

14

4

2

–

6

8

=

–

=

=

Lesson

3.4.5

Least Common Multiples

Guided Practice

Solve the following problems.

5.

6.

**TICKET PROBLEM

Solution follows…

5

2

1

1

3

3

1

8

2

4

6

4

2

4

7

9

3

9

9

9

8

2

10

14

10

11

13

18

13

12

15

23

25

25

+

+

44

25

12

15

18

15

22

63

1

9

5

35

18

12

1

1

4

+

–

–

+

–

–

5

3

4

Lesson

3.4.5

Least Common Multiples

Independent Practice

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

7. 8. 9.

10. 11. 12.

13. 14. 15.

–

Solution follows…

7

1

+

+

30

9

1

6

4

7

1

27

24

45

21

75

17

10

31

10

15

15

13

24

47

73

135

+

+

49

35

42

56

56

49

90

42

84

274

–

–

525

Lesson

3.4.5

Least Common Multiples

Independent Practice

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

26. 27. 28.

29. 30. 31.

Solution follows…

Lesson

3.4.5

Least Common Multiples

Round 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.