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HomeworkHomework Text page 190, #16-36 evensText page 190, #16-36 evens

16) 3616) 36 18) 1618) 16 20) 4220) 42 22) 16522) 165 24) 12 24) 12 26) 42026) 420 28) 80a³28) 80a³

30) 51b³30) 51b³ 32) 120s32) 120s 34) 200a²34) 200a² 36) 24 figures36) 24 figures

44

ObjectiveObjective

SWBAT find the least common SWBAT find the least common multiple of two numbers.multiple of two numbers.

Section 4.4 “Least Common Multiple”Section 4.4 “Least Common Multiple”

Least Common MultipleLeast Common Multiple

Of all the common multiples of two Of all the common multiples of two numbers, the smallest is the numbers, the smallest is the least least common multiplecommon multiple. .

A A multiplemultiple of a number is the of a number is the product of the number and any other product of the number and any other number greater than zero.number greater than zero.

What are the multiples of 5? What are the multiples of 5? Think of counting by fives…Think of counting by fives…

5:5: ,5 10, 15, 20, 25, 30, 35, 40, 45, 50, 55 … ,5 10, 15, 20, 25, 30, 35, 40, 45, 50, 55 …

Find the LCM of 180 and 378.Find the LCM of 180 and 378.

When the numbers are too large to list the multiplesWhen the numbers are too large to list the multiplesof the two numbers, find the prime factorization of each of the two numbers, find the prime factorization of each number. number. The LCM is the product of the common prime The LCM is the product of the common prime factors and the factors that are not commonfactors and the factors that are not common..

180180: : 22 · · 2 · 3 · 32 · 3 · 3 · · 55

378378: : 2 · 3 · 32 · 3 · 3 · · 3 · 73 · 7

From the list you can see that 2, 3, and 3 are common prime From the list you can see that 2, 3, and 3 are common prime Factors, and the factors that are not common are 2, 3, 5, and 7. Factors, and the factors that are not common are 2, 3, 5, and 7. The LCM is then The LCM is then 2 · 3 · 32 · 3 · 3 · · 2 · 52 · 5 · · 3 · 73 · 7 which is 3780. which is 3780.

Least Common MultipleLeast Common Multiple

Common Common FactorsFactors

Not Common Not Common FactorsFactors

2 · 3 · 32 · 3 · 3 2 2 · · 55 3 · 73 · 7

Find the LCM of 15, 30, and 50.Find the LCM of 15, 30, and 50.

When you have three or more numbers,When you have three or more numbers,the common prime factors may only be shared the common prime factors may only be shared by two of the numbers.by two of the numbers. The LCM is the product of the The LCM is the product of the common prime factors and the factors that are not commoncommon prime factors and the factors that are not common..

15: 15: 3 · 5 3 · 5

30: 30: 2 · 3 · 52 · 3 · 5

From the list you can see that 2, 3, and 5 are common prime From the list you can see that 2, 3, and 5 are common prime factors, and the factor that is not common is 5. factors, and the factor that is not common is 5. The LCM is then The LCM is then 2 · 3 · 52 · 3 · 5 · 5 · 5 which is 150. which is 150.

Least Common MultipleLeast Common Multiple

Common Common FactorsFactors

Not Common Not Common FactorsFactors

2 · 3 · 52 · 3 · 5

55

50: 50: 2 ·2 · 5 5 · 5 · 5

Find the LCM of Find the LCM of 10ab10ab and and 6b6b..The LCM is the product of the common prime The LCM is the product of the common prime factors and the factors that are not commonfactors and the factors that are not common..

10ab10ab: : 22 · · 5 · a5 · a · · bb

6b6b: : 22 · · 3 ·3 · bb

From the list you can see that 2 and b are common prime From the list you can see that 2 and b are common prime factors, and the factors that are not common are 3, 5, and a. factors, and the factors that are not common are 3, 5, and a. The LCM is then The LCM is then 2 · b2 · b · 5 · a · 5 · a · 3 · 3 which is 30ab. which is 30ab.

Least Common MultipleLeast Common Multiple

Common Common FactorsFactors

Not Common Not Common FactorsFactors

2 · b 2 · b 5 · a5 · a 3 3

Find the LCM of 2Find the LCM of 2a³b and 3aba³b and 3ab..The LCM is the product of the common prime The LCM is the product of the common prime factors and the factors that are not commonfactors and the factors that are not common..

22a³ba³b : : 2 ·2 · a a · a · a · · a · a · b b

3ab 3ab :: 3 · 3 · aa · · bb · b · b · b · b · b · b · b · b

From the list you can see that From the list you can see that a and ba and b are common prime are common prime factors, and the factors that are not common are 2, 3, a, a, b,factors, and the factors that are not common are 2, 3, a, a, b,b, b, and b. The LCM is then b, b, and b. The LCM is then a · ba · b · 2 · a · 2 · a · a · 3· a · 3 · b · b· b · b · b · b· b · b

which is which is 6a³b. 6a³b.

Least Common MultipleLeast Common Multiple

Common FactorsCommon Factors Not Common FactorsNot Common Factors

a · b a · b 2 · a · a 2 · a · a 3 · b · b · b · b3 · b · b · b · b

55

55

55

Least Common Least Common DenominatorDenominator

– The LCD of two or more fractions is the The LCD of two or more fractions is the least common multiple of the least common multiple of the denominators. Use the LCD to compare denominators. Use the LCD to compare and order fractions.and order fractions.

““Using the LCD” Using the LCD”

Compare the fractions and .Compare the fractions and .

List the equivalent fractions and find two with the List the equivalent fractions and find two with the same denominator. Then compare. same denominator. Then compare.

4

3

4

3same denominator, same denominator, now comparenow compare

3

2

3

2

8

6

12

9

6

4

9

6

12

8== == ==

====

Therefore, 3/4 Therefore, 3/4 isisgreater than greater than 2/3. 2/3.

Compare the Following Compare the Following FractionsFractions

4

3

6

2

12

9

4

3

12

4

6

2

4

3

6

2<

3

1

12

5

12

4

3

1

12

5

12

5

3

1

12

5>

7

2

4

1

28

8

7

2

28

7

4

1

7

2

4

1 <

Order the fractions Order the fractions from least to from least to

greatest.greatest. 15

7,9

4,4

3

Find the LCD for 4, 9, & 15. Find the LCD for 4, 9, & 15. LCD = 180

Rewrite equivalent fractions using the LCD. Rewrite equivalent fractions using the LCD.

4

3,

15

7,9

4

180

84

15

7

180

80

9

4

180

135

4

3

Order the fractions Order the fractions from least to from least to

greatest.greatest. 9

11,3

11,

6

7

Find the LCD for 6, 9, & 3. Find the LCD for 6, 9, & 3. LCD = 18

Rewrite equivalent fractions using the LCD. Rewrite equivalent fractions using the LCD.

3

11,

9

11,6

7

18

24

3

4

3

11

18

22

9

11

18

21

6

7

Rewrite the variable Rewrite the variable expressions with a expressions with a

common common denominator.denominator.

24

3,

5

2

abb

a

Find the LCD for 5b and 4ab². Find the LCD for 5b and 4ab². LCD = 20ab²

Rewrite equivalent fractions using the LCD. Rewrite equivalent fractions using the LCD.

22 20

?

4

3

abab

220

?

5

2

abb

a

222 20

15

54

53

4

3

ababab

2

2

20

8

45

42

5

2

ab

ba

abb

aba

b

a

Rewrite the variable Rewrite the variable expressions with a expressions with a

common common denominator.denominator.

xyy

x

5

2,

4

32

Find the LCD for 5xy and 4y². Find the LCD for 5xy and 4y². LCD = 20xy²

Rewrite equivalent fractions using the LCD. Rewrite equivalent fractions using the LCD.

220

?

5

2

xyxy

22 20

?

4

3

xyy

x 2

2

22 20

15

54

53

4

3

xy

x

xy

xx

y

x

220

8

45

42

5

2

xy

y

yxy

y

xy

Rewrite the variable Rewrite the variable expressions with a expressions with a

common common denominator.denominator.

2

2

42

3

9

5,

7

2

s

tr

tr

s

422

5

63

18

tsr

s422

54

63

35

tsr

tr

CHALLENGE

CHALLENGE

HomeworkHomework

Text page 190, #40-68 evensText page 190, #40-68 evens