1 5 the least common multiple

The Least Common Multiple (LCM)
Frank Ma © 2011

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,

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

A common multipleof 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,

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 method
II. The construction method

Methods of Finding LCM
Searching Method

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

Searching Method
Example A. Find the LCM of 18, 24, 16.

Searching Method
List the multiples of 24:

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

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

Searching Method
so the LCM of {18, 24, 16} is 144.

Searching Method
so the LCM of {18, 24, 16} is 144.
Construction Method:

Searching Method
so the LCM of {18, 24, 16} is 144.
Construction Method: Given two or more quantities,
I. factor each quantity completely,

Searching Method
so the LCM of {18, 24, 16} is 144.
II. multiply the highest power of each factor that appears in the factorization to get the LCM.

Searching Method
so the LCM of {18, 24, 16} is 144.
Example B. a. Construct the LCM of 18, 24, 20.

Searching Method
so the LCM of {18, 24, 16} is 144.
Factor each number completely

Searching Method
so the LCM of {18, 24, 16} is 144.
18 = 2*32

Searching Method
so the LCM of {18, 24, 16} is 144.
18 = 2*32
24 = 233

Searching Method
so the LCM of {18, 24, 16} is 144.
18 = 2*32
24 = 233
20 = 225

Searching Method
so the LCM of {18, 24, 16} is 144.
18 = 2*32
24 = 233
20 = 225
Take the highest of the each factor,

Searching Method
so the LCM of {18, 24, 16} is 144.
18 = 2*32
24 = 233
20 = 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

Factor each quantity.

x2 – 3x + 2 = (x – 2)(x – 1)

x2 – 3x + 2 = (x – 2)(x – 1)
x2 + x – 2 = (x + 2)(x – 1)

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 =

x2 – 3x + 2 = (x – 2)(x – 1)
x2 + x – 2 = (x + 2)(x – 1)
LCM = (x – 2)

x2 – 3x + 2 = (x – 2)(x – 1)
x2 + x – 2 = (x + 2)(x – 1)
LCM = (x – 2)(x – 1)

x2 – 3x + 2 = (x – 2)(x – 1)
x2 + x – 2 = (x + 2)(x – 1)
LCM = (x – 2)(x – 1)(x +2)

x2 – 3x + 2 = (x – 2)(x – 1)
x2 + x – 2 = (x + 2)(x – 1)
LCM = (x – 2)(x – 1)(x +2)
d. Construct the LCM of x4 – 4x2, x2 – 4x +4

x2 – 3x + 2 = (x – 2)(x – 1)
x2 + x – 2 = (x + 2)(x – 1)
LCM = (x – 2)(x – 1)(x +2)

x2 – 3x + 2 = (x – 2)(x – 1)
x2 + x – 2 = (x + 2)(x – 1)
LCM = (x – 2)(x – 1)(x +2)
x4 – 4x2 = x2(x2 – 4)

x2 – 3x + 2 = (x – 2)(x – 1)
x2 + x – 2 = (x + 2)(x – 1)
LCM = (x – 2)(x – 1)(x +2)
x4 – 4x2 = x2(x2 – 4) = x2(x + 2)(x – 2)

x2 – 3x + 2 = (x – 2)(x – 1)
x2 + x – 2 = (x + 2)(x – 1)
LCM = (x – 2)(x – 1)(x +2)
x4 – 4x2 = x2(x2 – 4) = x2(x + 2)(x – 2)
x2 – 4x + 4 = (x – 2)2

x2 – 3x + 2 = (x – 2)(x – 1)
x2 + x – 2 = (x + 2)(x – 1)
LCM = (x – 2)(x – 1)(x +2)
x4 – 4x2 = x2(x2 – 4) = x2(x + 2)(x – 2)
x2 – 4x + 4 = (x – 2)2
LCM =

x2 – 3x + 2 = (x – 2)(x – 1)
x2 + x – 2 = (x + 2)(x – 1)
LCM = (x – 2)(x – 1)(x +2)
x4 – 4x2 = x2(x2 – 4) = x2(x + 2)(x – 2)
x2 – 4x + 4 = (x – 2)2
LCM = x2

x2 – 3x + 2 = (x – 2)(x – 1)
x2 + x – 2 = (x + 2)(x – 1)
LCM = (x – 2)(x – 1)(x +2)
x4 – 4x2 = x2(x2 – 4) = x2(x + 2)(x – 2)
x2 – 4x + 4 = (x – 2)2
LCM = x2(x – 2)2

x2 – 3x + 2 = (x – 2)(x – 1)
x2 + x – 2 = (x + 2)(x – 1)
LCM = (x – 2)(x – 1)(x +2)
x4 – 4x2 = x2(x2 – 4) = x2(x + 2)(x – 2)
x2 – 4x + 4 = (x – 2)2
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 .

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

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.
xy3, xy
x2y, xy3
xy3, xy
3.
1.
2.
4xy3, 6xy
x2y, xy3, xy2
9xy3, 12x3y, 15x2y3
4.
5.
6.
(x + 2), (x – 1)
7.
(x + 2), (2x – 4)
8.
(4x + 12), (10 – 5x)
9.
(9x + 18), (2x + 4)
(–x + 2), (4x – 8)
(4x + 6), (2x + 3)
10.
11.
12.
x2(x + 2), (x + 2)
(x – 1)(x + 2), (x + 2)(x – 2)
13.
14.
(3x + 6)(x – 2), (x + 2)(x + 1)
(x – 3)(2x + 1), (2x + 1)(x – 2)
15.
16.
(x – 3)(x – 3), (3 – x)(x + 2)
x2(x – 2), (x – 2)x
17.
18.
x2(x + 2), (x + 2)(–x – 2)
(x – 1)(1 – x), (–1 – x)(x + 1)
19.
20.
x2 – 4x + 4, x2 – 4
21.
x2 – x – 6, x2 – x – 2
22.
x2 – x – 6, x2 – 9
23.
x2 + 5x – 6, x2 – x – 6
24.
x2 + 2x – 3, 1 – x2
25.
–x2 – 2x + 3, x2 – x
26.
2x2 + 5x – 3, x – 4x3
27.
x2 – 5x + 4, x3 – 16x
28.

1 5 the least common multiple

by math123b on Aug 28, 2011

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