14 the lcm and the multiplier method for addition and subtraction of rational expressions

The Least Common Multiple (LCM)

Frank Ma © 2011

The Least Common Multiple (LCM)We say m is a multiple of x if m can be divided by x.

For example, 12 is a multiple of 2


For example, 12 is a multiple of 2. 12 is also a multiple of 3, 4, 6 or 12 itself.




xy2 is a multiple of x,

We say m is a multiple of x if m can be divided by x.



xy2 is a multiple of x, xy2 is also a multiple of y or xy.




A common multiple of two or more quantities is a multiple of all the given quantities.





A common multiple of two or more quantities is a multiple of all the given quantities. For example, 24 is a common multiple of 4 and 6,







xy2 is a common multiple of x and y.


The least common multiple (LCM) of two or more numbers is the smallest common multiple of all the given numbers.









For example, 12 is the LCM of 4 and 6.








For example, 12 is the LCM of 4 and 6, xy is the LCM of x and y,








For example, 12 is the LCM of 4 and 6, xy is the LCM of x and y, and that 12xy is the LCM of 4x and 6y.











We give two methods for finding the LCM.I. The searching method









We give two methods for finding the LCM.I. The searching methodII. The construction method




Searching Method

Methods of Finding LCM

Searching Method Given a list of quantities, list the multiples of the largest quantity and test them one by one until we find the LCM.


Searching Method Given a list of quantities, list the multiples of the largest quantity and test them one by one until we find the LCM. Example A. Find the LCM of 18, 24, 16.



List the multiples of 24:



List the multiples of 24: 24, 48, 72,



List the multiples of 24: 24, 48, 72, 96, 120, 144, …



List the multiples of 24: 24, 48, 72, 96, 120, 144, …so the LCM of {18, 24, 16} is 144.





Construction Method:




Construction Method: Given two or more quantities,I. factor each quantity completely,




Construction Method: Given two or more quantities,I. factor each quantity completely,II. multiply the highest power of each factor that appears in the factorization to get the LCM.




Example B. a. Construct the LCM of 18, 24, 20.






Factor each number completely






Factor each number completely 18 = 2*32







24 = 233







24 = 23320 = 225







24 = 23320 = 225

Take the highest of the each factor,







24 = 23320 = 225

Take the highest of the each factor,so the LCM is 23325 = 360.


b. Construct the LCM of x2y3z, x3yz4, x3zw



All quantities are factored, Take the highest power of each factor.



All quantities are factored, Take the highest power of each factor. The LCM =



All quantities are factored, Take the highest power of each factor. The LCM = x3



All quantities are factored, Take the highest power of each factor. The LCM = x3y3



All quantities are factored, Take the highest power of each factor. The LCM = x3y3z4



All quantities are factored, Take the highest power of each factor. The LCM = x3y3z4w.





c. Construct the LCM of x2 – 3x + 2, x2 – x – 2




c. Construct the LCM of x2 – 3x + 2, x2 – x – 2

Factor each quantity.




c. Construct the LCM of x2 – 3x + 2, x2 + x – 2

Factor each quantity.x2 – 3x + 2 = (x – 2)(x – 1)





Factor each quantity.x2 – 3x + 2 = (x – 2)(x – 1) x2 + x – 2 = (x + 2)(x – 1)





Factor each quantity.x2 – 3x + 2 = (x – 2)(x – 1) x2 + x – 2 = (x + 2)(x – 1) Take the highest power of each factor to get the LCM =





Factor each quantity.x2 – 3x + 2 = (x – 2)(x – 1) x2 + x – 2 = (x + 2)(x – 1) Take the highest power of each factor to get the LCM = (x – 2)





Factor each quantity.x2 – 3x + 2 = (x – 2)(x – 1) x2 + x – 2 = (x + 2)(x – 1) Take the highest power of each factor to get the LCM = (x – 2)(x – 1)





Factor each quantity.x2 – 3x + 2 = (x – 2)(x – 1) x2 + x – 2 = (x + 2)(x – 1) Take the highest power of each factor to get the LCM = (x – 2)(x – 1)(x +2)




d. Construct the LCM of x4 – 4x2, x2 – 4x +4







Factor each quantity.







Factor each quantity.x4 – 4x2 = x2(x2 – 4)







Factor each quantity.x4 – 4x2 = x2(x2 – 4) = x2(x + 2)(x – 2)







Factor each quantity.x4 – 4x2 = x2(x2 – 4) = x2(x + 2)(x – 2) x2 – 4x + 4 = (x – 2)2







Factor each quantity.x4 – 4x2 = x2(x2 – 4) = x2(x + 2)(x – 2) x2 – 4x + 4 = (x – 2)2 Take the highest power of each factor to get the LCM =







Factor each quantity.x4 – 4x2 = x2(x2 – 4) = x2(x + 2)(x – 2) x2 – 4x + 4 = (x – 2)2 Take the highest power of each factor to get the LCM = x2







Factor each quantity.x4 – 4x2 = x2(x2 – 4) = x2(x + 2)(x – 2) x2 – 4x + 4 = (x – 2)2 Take the highest power of each factor to get the LCM = x2(x – 2)2







Factor each quantity.x4 – 4x2 = x2(x2 – 4) = x2(x + 2)(x – 2) x2 – 4x + 4 = (x – 2)2 Take the highest power of each factor to get the LCM = x2(x – 2)2(x + 2)



The Construction Method takes just enough of each factor to make the LCM which "covers" all the quantities on the list.


The Construction Method takes just enough of each factor to make the LCM which "covers" all the quantities on the list. The following is an example that illustrates the principle behind this method .


Methods of Finding LCMThe Construction Method takes just enough of each factor to make the LCM which "covers" all the quantities on the list. The following is an example that illustrates the principle behind this method .

Example C. A student wants to take enough courses so she meets the requirement to apply to colleges A, B, and C. The following is a table of requirements for each college, how many years of each subject does she need?

Science Math. English Arts

A 2 yrs 3 yrs 3 yrs

B 3 yrs 3 yrs 2 yrs

C 2 yrs 2 yrs 4 yrs 1 yrs




A 2 yrs 3 yrs 3 yrs

B 3 yrs 3 yrs 2 yrs


Science-3 yr Math-3yrs English-4 yrs Arts-1yr




A 2 yrs 3 yrs 3 yrs

B 3 yrs 3 yrs 2 yrs


Science-3 yr Math-3yrs English-4 yrs Arts-1yr



So we take the highest power of each factor to make the LCM.

Ex. A. Find the LCM of the given expressions. Methods of Finding LCM

xy3, xy 1. x2y, xy3 2. xy3, xyz 3.

4xy3, 6xy 4. x2y, xy3, xy2 5. 9xy3, 12x3y, 15x2y3 6.

(x + 2), (x – 1) 7. (x + 2), (2x – 4) 8. (4x + 12), (10 – 5x) 9.

(4x + 6), (2x + 3) 10. (9x + 18), (2x + 4)

11. (–x + 2), (4x – 8) 12.

x2(x + 2), (x + 2) 13. (x – 1)(x + 2), (x + 2)(x – 2)

14.

(x – 3)(x – 3), (3 – x)(x + 2) 17. x2(x – 2), (x – 2)x 18.

19. x2(x + 2), (x + 2)(–x – 2) 20. (x – 1)(1 – x), (–1 – x)(x + 1)

(3x + 6)(x – 2), (x + 2)(x + 1) 15. (x – 3)(2x + 1), (2x + 1)(x – 2)

16.

x2 – 4x + 4, x2 – 4 21. x2 – x – 6, x2 – x – 2 22.

x2 + 2x – 3, 1 – x225. –x2 – 2x + 3, x2 – x 26.

x2 – x – 6, x2 – 923. x2 + 5x – 6, x2 – x – 6 24.

2x2 + 5x – 3, x – 4x3 27. x2 – 5x + 4, x3 – 16x 28.

14 the lcm and the multiplier method for addition and subtraction of rational expressions

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