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LCM - least common multiple

by LZ
(NJ)











































Least Common Multiple (LCM)

Use the concept of the Least Common Multiple (LCM) to answer the following question:

Casey's watch beeps every 15 minutes.

Her brother's watch beeps every 10 minutes.

The last time both watches beeped was 5:30 pm.


What time will it be the next time both watches beep at the same time?

Comments for LCM - least common multiple

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Jan 09, 2014
watch alarms beep at same time
by: Staff


Answer

Part I

The LCM (least common multiple) of two or more numbers is the smallest number that is divisible by each one of the given numbers.

To understand the LCM, begin by calculating the multiples of each number.

For the watch that beeps every 10 minutes, the multiples are:

1 * 10 = 10

2 * 10 = 20

3 * 10 = 30

4 * 10 = 40

5 * 10 = 50

multiples of 10 are: 10, 20, 30, 40, 50

For the watch that beeps every 15 minutes, the multiples are:

1 * 15 = 15

2 * 15 = 30

3 * 15 = 45

4 * 15 = 60

5 * 15 = 75

multiples of 15 are: 15, 30, 45, 60, 75




multiples of 10






multiples of 15





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Jan 09, 2014
watch alarms beep at same time
by: Staff


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Part II



the lowest common multiple of 10 and 15 is the smallest multiple that is common to each:

multiples of 10 are: 10, 20, 30, 40, 50

multiples of 15 are: 15, 30, 45, 60, 75

the lowest common multiple of 10 and 15 = 30

that means that every 30 minutes, both watches will beep together

both watches beep at 5:30 pm + 30 minutes = 6:00 pm




lowest common multiple of 10 and 15





The LCM of 10 and 15 is 30.  This means that both watches will beep together every 30 minutes








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Jan 09, 2014
watch alarms beep at same time
by: Staff


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Part III



Both alarms will sound simultaneously at 6:00 pm





the final answer to your question is:

the next time both watches beep together will be 6:00 pm



 Final Answer:  the next time both alarms will sound together is 6:00 pm







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Jan 09, 2014
watch alarms beep at same time
by: Staff

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Part IV


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Although you now have the answer to your question, the method we used to determine the answer is cumbersome.

Listing each multiple for each number requires a lot of unnecessary work. The amount of work will be even greater if there are more than two numbers involved.

For example, suppose you needed to compute the LCM for three watches: one that beeped every 10 minutes, one that beeped every 12 minutes, and one that beeped every 15 minutes.

using the method we employed above for the two watches:

For the watch that beeps every 10 minutes, the multiples are:

1 * 10 = 10

2 * 10 = 20

3 * 10 = 30

4 * 10 = 40

5 * 10 = 50

6 * 10 = 60

7 * 10 = 70

8 * 10 = 80

multiples of 10 are: 10, 20, 30, 40, 50, 60, 70, 80

For the watch that beeps every 12 minutes, the multiples are:

1 * 12 = 12

2 * 12 = 24

3 * 12 = 36

4 * 12 = 48

5 * 12 = 60

6 * 12 = 72

7 * 12 = 84

8 * 12 = 96

multiples of 12 are: 12, 24, 36, 48, 60, 72, 84, 96


For the watch that beeps every 15 minutes, the multiples are:

1 * 15 = 15

2 * 15 = 30

3 * 15 = 45

4 * 15 = 60

5 * 15 = 75

6 * 15 = 90

7 * 15 = 105

8 * 15 = 120

multiples of 15 are: 15, 30, 45, 60, 75, 90, 105, 120


the lowest common multiple of 10, 12, and 15 is the smallest multiple that is common to each:

multiples of 10 are: 10, 20, 30, 40, 50, 60, 70, 80

multiples of 12 are: 12, 24, 36, 48, 60, 72, 84, 96

multiples of 15 are: 15, 30, 45, 60, 75, 90, 105, 120


the lowest common multiple of 10, 12, and 15 = 60

that means that every 60 minutes, all three watches will beep at the same time

the three watches will beep at 5:30 pm + 60 minutes = 6:30 pm



As you can see, listing each multiple for 3 numbers is even more time consuming than listing each multiple for 2 numbers.


prime factorization

The LCM can be quickly determined using prime factorization.

The LCM is the product of multiplying the highest power of each prime number together.

The prime factors of the three numbers in our example are:

10 = 2¹ * 5¹

12 = 2 * 2 * 3 = 2² * 3¹

15 = 3¹ * 5¹


To compute the LCM:

Choose the 2 with the highest exponent: 2²

Choose the 3 with the highest exponent: 3¹

Choose the 5 with the highest exponent: 5¹



Multiply all three of these numbers together

LCM (10,12,15) = 2² * 3¹ * 5¹ = 2² * 3¹ * 5¹ = 60

>>> LCM (least common multiple) for the values 10, 12, and 15 = 60




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Jan 09, 2014
watch alarms beep at same time
by: Staff

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Part V




The most common application of the LCM is adding and subtracting fractions with unlike denominators.

For example, add the following two fractions using LCM to find the least common denominator:

3/4 + 1/84 = ?

The prime factors of the two denominators in the example are:
4 = 2²

84 = 2² * 3¹ * 7¹

To compute the LCM:

Choose the 2 with the highest exponent: 2²

Choose the 3 with the highest exponent: 3¹

Choose the 7 with the highest exponent: 7¹



Multiply all three of these numbers together

LCM (4, 84) = 2² * 3¹ * 7¹ = 84

>>> LCM (least common multiple) for the denominators 4 and 84 = 84
add the fractions

3/4 + 1/84

= (3/4)*(21/21) + (1/84)

= (63/84) + (4/84)

= (67/84)

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other practical uses of LCM

whenever you have an event which repeats itself, the LCM can be used

There are 6 hot dog buns in a package, but 8 hot dogs in a package. How many packages of buns, and how many packages of hot dogs should you buy to ensure there is one lot dog for every bun, and vice versa?

6 = 2¹ * 3¹

8 = 2³

LCM (6, 8) = 2³ * 3¹ = 24

you must purchase 24 hot dog buns and 24 hot dogs

4 packages of hot dog buns * 6 buns per package = 24 buns

3 packages of hot dogs * 8 hot dogs per = 24 hot dogs

you must purchase 4 packages of hot dog buns and 3 packages of hot dogs


Mike has golf lessons every fifth day and swimming lessons every third day. If he had both a golf lesson and a swimming lesson today, how many days will it be before he has both lessons on the same day, again.

5 = 5¹

3 = 3¹

LCM (5, 3) = 5¹ * 3¹ = 15

both lessons will fall upon the same day again in 15 days


Boxes that are 15 inches tall are being stacked next to boxes that are 18 inches tall. What is the shortest height at which the two stacks will be the same height?


15 = 3¹ * 5¹

18 = 3² * 2¹

LCM (5, 3) = 2¹ * 3² * 5¹ = 90

the height of both stacks will be equal when the height of each stack is 90 inches

A homeowner wants to line up three rows of tile around the wall near the baseboard in one of the rooms. The length of the tile he will use in the first row is 5 inches. The length of the tile he will use in the second row is 3 inches. The length of the tile he will use in the third row is 2 inches. How long will the three rows need to be so the total length of the tiles in each row is the same?

5 = 5¹

3 = 3¹

2 = 2¹

LCM (5, 3, 2) = 2¹ * 3¹ * 5¹ = 30

the length of each row will be the same every 30 inches


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Jan 09, 2014
watch alarms beep at same time
by: Staff

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Part VI



A school teacher is arranging the seating chart so that there are an equal number of A, B, C, and D students in each row. There are 6 "A" students, 12 "B" students, 12 "C" students, and 6 "D" students in the class. How many students should be assigned seats in each row?

6 = 3¹ * 2¹

12 = 3¹ * 2²

12 = 3¹ * 2²

6 = 3¹ * 2¹

LCM (9, 15, 25, 10) = 2² * 3¹ = 12

The length of each row will be 12 students. There will be 3 rows (since there are 36 students in the class). Each row will seat 2 "A" students, 4 "B" students, 4 "C" students, and 2 "D" students.





Thanks for writing.

Staff
www.solving-math-problems.com


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