Uploaded byalg1testreview
PPTX, PDF518 views

54 the least common multiple

The document discusses methods for finding the least common multiple (LCM) of numbers. It defines a multiple as a number that can be divided evenly by another number. The LCM is the smallest number that is a multiple of all numbers given. Two methods are described: the searching method which tests multiples of the largest number, and the construction method which factors each number and multiplies the highest powers of common factors. Examples are provided to illustrate both methods.

Embed presentation

Download to read offline
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 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. 12 is also a multiple of 3, 4, 6 or 12 itself. 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. 12 is also a multiple of 3, 4, 6 or 12 itself. The Least Common Multiple (LCM) xy2 is a multiple of x, We say m is a multiple of x if m can be divided by x.
For example, 12 is a multiple of 2. 12 is also a multiple of 3, 4, 6 or 12 itself. The Least Common Multiple (LCM) xy2 is a multiple of x, xy2 is also a multiple of y or xy. We say m is a multiple of x if m can be divided by x.
For example, 12 is a multiple of 2. 12 is also a multiple of 3, 4, 6 or 12 itself. The Least Common Multiple (LCM) A common multiple of two or more quantities is a multiple of all the given quantities. xy2 is a multiple of x, xy2 is also a multiple of y or xy. We say m is a multiple of x if m can be divided by x.
For example, 12 is a multiple of 2. 12 is also a multiple of 3, 4, 6 or 12 itself. The Least Common Multiple (LCM) 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 multiple of x, xy2 is also a multiple of y or xy. We say m is a multiple of x if m can be divided by x.
For example, 12 is a multiple of 2. 12 is also a multiple of 3, 4, 6 or 12 itself. The Least Common Multiple (LCM) 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 multiple of x, xy2 is also a multiple of y or xy. xy2 is a common multiple of x and y. We say m is a multiple of x if m can be divided by x.
The least common multiple (LCM) of two or more numbers is the smallest common multiple of all the given numbers. For example, 12 is a multiple of 2. 12 is also a multiple of 3, 4, 6 or 12 itself. The Least Common Multiple (LCM) 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 multiple of x, xy2 is also a multiple of y or xy. xy2 is a common multiple of x and y. We say m is a multiple of x if m can be divided by x.
The least common multiple (LCM) of two or more numbers is the smallest common multiple of all the given numbers. For example, 12 is a multiple of 2. 12 is also a multiple of 3, 4, 6 or 12 itself. For example, 12 is the LCM of 4 and 6. The Least Common Multiple (LCM) 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 multiple of x, xy2 is also a multiple of y or xy. xy2 is a common multiple of x and y. We say m is a multiple of x if m can be divided by x.
The least common multiple (LCM) of two or more numbers is the smallest common multiple of all the given numbers. For example, 12 is a multiple of 2. 12 is also a multiple of 3, 4, 6 or 12 itself. For example, 12 is the LCM of 4 and 6, xy is the LCM of x and y, The Least Common Multiple (LCM) 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 multiple of x, xy2 is also a multiple of y or xy. xy2 is a common multiple of x and y. We say m is a multiple of x if m can be divided by x.
The least common multiple (LCM) of two or more numbers is the smallest common multiple of all the given numbers. For example, 12 is a multiple of 2. 12 is also a multiple of 3, 4, 6 or 12 itself. 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. The Least Common Multiple (LCM) 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 multiple of x, xy2 is also a multiple of y or xy. xy2 is a common multiple of x and y. We say m is a multiple of x if m can be divided by x.
The least common multiple (LCM) of two or more numbers is the smallest common multiple of all the given numbers. For example, 12 is a multiple of 2. 12 is also a multiple of 3, 4, 6 or 12 itself. 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. The Least Common Multiple (LCM) 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, We give two methods for finding the LCM. I. The searching method xy2 is a multiple of x, xy2 is also a multiple of y or xy. xy2 is a common multiple of x and y. We say m is a multiple of x if m can be divided by x.
The least common multiple (LCM) of two or more numbers is the smallest common multiple of all the given numbers. For example, 12 is a multiple of 2. 12 is also a multiple of 3, 4, 6 or 12 itself. 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. The Least Common Multiple (LCM) 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, We give two methods for finding the LCM. I. The searching method II. The construction method xy2 is a multiple of x, xy2 is also a multiple of y or xy. xy2 is a common multiple of x and y. We say m is a multiple of x if m can be divided by x.
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. 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. Example A. Find the LCM of 18, 24, 16. 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. Example A. Find the LCM of 18, 24, 16. List the multiples of 24: 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. Example A. Find the LCM of 18, 24, 16. List the multiples of 24: 24, 48, 72, 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. Example A. Find the LCM of 18, 24, 16. List the multiples of 24: 24, 48, 72, 96, 120, 144, … 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. Example A. Find the LCM of 18, 24, 16. List the multiples of 24: 24, 48, 72, 96, 120, 144, … so the LCM of {18, 24, 16} is 144. 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. Example A. Find the LCM of 18, 24, 16. List the multiples of 24: 24, 48, 72, 96, 120, 144, … so the LCM of {18, 24, 16} is 144. Methods of Finding LCM Construction Method:
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: 24, 48, 72, 96, 120, 144, … so the LCM of {18, 24, 16} is 144. Methods of Finding LCM Construction Method: Given two or more quantities, I. factor each quantity completely,
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: 24, 48, 72, 96, 120, 144, … so the LCM of {18, 24, 16} is 144. Methods of Finding LCM 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.
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: 24, 48, 72, 96, 120, 144, … so the LCM of {18, 24, 16} is 144. Methods of Finding LCM Example B. a. Construct the LCM of 18, 24, 20. 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.
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: 24, 48, 72, 96, 120, 144, … so the LCM of {18, 24, 16} is 144. Methods of Finding LCM Example B. a. Construct the LCM of 18, 24, 20. Factor each number 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.
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: 24, 48, 72, 96, 120, 144, … so the LCM of {18, 24, 16} is 144. Methods of Finding LCM Example B. a. Construct the LCM of 18, 24, 20. Factor each number completely 18 = 2*32 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.
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: 24, 48, 72, 96, 120, 144, … so the LCM of {18, 24, 16} is 144. Methods of Finding LCM Example B. a. Construct the LCM of 18, 24, 20. Factor each number completely 18 = 2*32 24 = 233 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.
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: 24, 48, 72, 96, 120, 144, … so the LCM of {18, 24, 16} is 144. Methods of Finding LCM Example B. a. Construct the LCM of 18, 24, 20. Factor each number completely 18 = 2*32 24 = 233 20 = 225 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.
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: 24, 48, 72, 96, 120, 144, … so the LCM of {18, 24, 16} is 144. Methods of Finding LCM Example B. a. Construct the LCM of 18, 24, 20. Factor each number completely 18 = 2*32 24 = 233 20 = 225 Take the highest of the each factor, 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.
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: 24, 48, 72, 96, 120, 144, … so the LCM of {18, 24, 16} is 144. Methods of Finding LCM Example B. a. Construct the LCM of 18, 24, 20. Factor each number completely 18 = 2*32 24 = 233 20 = 225 Take the highest of the each factor, 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.
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: 24, 48, 72, 96, 120, 144, … so the LCM of {18, 24, 16} is 144. Methods of Finding LCM Example B. a. Construct the LCM of 18, 24, 20. Factor each number completely 18 = 2*32 24 = 233 20 = 225 Take the highest of the each factor, 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.
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: 24, 48, 72, 96, 120, 144, … so the LCM of {18, 24, 16} is 144. Methods of Finding LCM Example B. a. Construct the LCM of 18, 24, 20. Factor each number completely 18 = 2*32 24 = 233 20 = 225 Take the highest of the each factor, 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.
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: 24, 48, 72, 96, 120, 144, … so the LCM of {18, 24, 16} is 144. Methods of Finding LCM Example B. a. Construct the LCM of 18, 24, 20. Factor each number completely 18 = 2*32 24 = 233 20 = 225 Take the highest of the each factor, so the LCM is 23325 = 360. 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.
b. Construct the LCM of x2y3z, x3yz4, x3zw Methods of Finding LCM
b. Construct the LCM of x2y3z, x3yz4, x3zw All quantities are factored, Take the highest power of each factor. Methods of Finding LCM
b. Construct the LCM of x2y3z, x3yz4, x3zw All quantities are factored, Take the highest power of each factor. The LCM = Methods of Finding LCM
b. Construct the LCM of x2y3z, x3yz4, x3zw All quantities are factored, Take the highest power of each factor. The LCM = x3 Methods of Finding LCM
b. Construct the LCM of x2y3z, x3yz4, x3zw All quantities are factored, Take the highest power of each factor. The LCM = x3y3 Methods of Finding LCM
b. Construct the LCM of x2y3z, x3yz4, x3zw All quantities are factored, Take the highest power of each factor. The LCM = x3y3z4 Methods of Finding LCM
b. Construct the LCM of x2y3z, x3yz4, x3zw All quantities are factored, Take the highest power of each factor. The LCM = x3y3z4w. Methods of Finding LCM
b. Construct the LCM of x2y3z, x3yz4, x3zw All quantities are factored, Take the highest power of each factor. The LCM = x3y3z4w. Methods of Finding LCM c. Construct the LCM of x2 – 3x + 2, x2 – x – 2
b. Construct the LCM of x2y3z, x3yz4, x3zw All quantities are factored, Take the highest power of each factor. The LCM = x3y3z4w. Methods of Finding LCM c. Construct the LCM of x2 – 3x + 2, x2 – x – 2 Factor each quantity.
b. Construct the LCM of x2y3z, x3yz4, x3zw All quantities are factored, Take the highest power of each factor. The LCM = x3y3z4w. Methods of Finding LCM c. Construct the LCM of x2 – 3x + 2, x2 + x – 2 Factor each quantity. x2 – 3x + 2 = (x – 2)(x – 1)
b. Construct the LCM of x2y3z, x3yz4, x3zw All quantities are factored, Take the highest power of each factor. The LCM = x3y3z4w. Methods of Finding LCM c. Construct the LCM of x2 – 3x + 2, x2 + x – 2 Factor each quantity. x2 – 3x + 2 = (x – 2)(x – 1) x2 + x – 2 = (x + 2)(x – 1)
b. Construct the LCM of x2y3z, x3yz4, x3zw All quantities are factored, Take the highest power of each factor. The LCM = x3y3z4w. Methods of Finding LCM c. Construct the LCM of x2 – 3x + 2, x2 + 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 =
b. Construct the LCM of x2y3z, x3yz4, x3zw All quantities are factored, Take the highest power of each factor. The LCM = x3y3z4w. Methods of Finding LCM c. Construct the LCM of x2 – 3x + 2, x2 + 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)
b. Construct the LCM of x2y3z, x3yz4, x3zw All quantities are factored, Take the highest power of each factor. The LCM = x3y3z4w. Methods of Finding LCM c. Construct the LCM of x2 – 3x + 2, x2 + 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)
b. Construct the LCM of x2y3z, x3yz4, x3zw All quantities are factored, Take the highest power of each factor. The LCM = x3y3z4w. Methods of Finding LCM c. Construct the LCM of x2 – 3x + 2, x2 + 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)(x +2)
b. Construct the LCM of x2y3z, x3yz4, x3zw All quantities are factored, Take the highest power of each factor. The LCM = x3y3z4w. Methods of Finding LCM d. Construct the LCM of x4 – 4x2, x2 – 4x +4 c. Construct the LCM of x2 – 3x + 2, x2 + 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)(x +2)
b. Construct the LCM of x2y3z, x3yz4, x3zw All quantities are factored, Take the highest power of each factor. The LCM = x3y3z4w. Methods of Finding LCM d. Construct the LCM of x4 – 4x2, x2 – 4x +4 Factor each quantity. c. Construct the LCM of x2 – 3x + 2, x2 + 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)(x +2)
b. Construct the LCM of x2y3z, x3yz4, x3zw All quantities are factored, Take the highest power of each factor. The LCM = x3y3z4w. Methods of Finding LCM d. Construct the LCM of x4 – 4x2, x2 – 4x +4 Factor each quantity. x4 – 4x2 = x2(x2 – 4) c. Construct the LCM of x2 – 3x + 2, x2 + 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)(x +2)
b. Construct the LCM of x2y3z, x3yz4, x3zw All quantities are factored, Take the highest power of each factor. The LCM = x3y3z4w. Methods of Finding LCM d. Construct the LCM of x4 – 4x2, x2 – 4x +4 Factor each quantity. x4 – 4x2 = x2(x2 – 4) = x2(x + 2)(x – 2) c. Construct the LCM of x2 – 3x + 2, x2 + 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)(x +2)
b. Construct the LCM of x2y3z, x3yz4, x3zw All quantities are factored, Take the highest power of each factor. The LCM = x3y3z4w. Methods of Finding LCM d. Construct the LCM of x4 – 4x2, x2 – 4x +4 Factor each quantity. x4 – 4x2 = x2(x2 – 4) = x2(x + 2)(x – 2) x2 – 4x + 4 = (x – 2)2 c. Construct the LCM of x2 – 3x + 2, x2 + 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)(x +2)
b. Construct the LCM of x2y3z, x3yz4, x3zw All quantities are factored, Take the highest power of each factor. The LCM = x3y3z4w. Methods of Finding LCM d. Construct the LCM of x4 – 4x2, x2 – 4x +4 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 = c. Construct the LCM of x2 – 3x + 2, x2 + 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)(x +2)
b. Construct the LCM of x2y3z, x3yz4, x3zw All quantities are factored, Take the highest power of each factor. The LCM = x3y3z4w. Methods of Finding LCM d. Construct the LCM of x4 – 4x2, x2 – 4x +4 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 c. Construct the LCM of x2 – 3x + 2, x2 + 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)(x +2)
b. Construct the LCM of x2y3z, x3yz4, x3zw All quantities are factored, Take the highest power of each factor. The LCM = x3y3z4w. Methods of Finding LCM d. Construct the LCM of x4 – 4x2, x2 – 4x +4 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 c. Construct the LCM of x2 – 3x + 2, x2 + 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)(x +2)
b. Construct the LCM of x2y3z, x3yz4, x3zw All quantities are factored, Take the highest power of each factor. The LCM = x3y3z4w. Methods of Finding LCM d. Construct the LCM of x4 – 4x2, x2 – 4x +4 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) c. Construct the LCM of x2 – 3x + 2, x2 + 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)(x +2)
The Construction Method takes just enough of each factor to make the LCM which "covers" all the quantities on the list. Methods of Finding LCM
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 LCM
Methods of Finding LCM 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 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 Methods of Finding LCM 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 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 Methods of Finding LCM 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 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 Science-3 yr Math-3yrs English-4 yrs Arts-1yr Methods of Finding LCM 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 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 Science-3 yr Math-3yrs English-4 yrs Arts-1yr Methods of Finding LCM 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? 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, xy1. x2y, xy32. xy3, xyz3. 4xy3, 6xy4. x2y, xy3, xy25. 9xy3, 12x3y, 15x2y36. (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)x18. 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 – 421. x2 – x – 6, x2 – x – 222. x2 + 2x – 3, 1 – x225. –x2 – 2x + 3, x2 – x26. x2 – x – 6, x2 – 923. x2 + 5x – 6, x2 – x – 624. 2x2 + 5x – 3, x – 4x327. x2 – 5x + 4, x3 – 16x28.

More Related Content

PPTX
1 f4 lcm and lcd
bymath123a
 
PPTX
14 lcm, addition and subtraction of fractions
byalg1testreview
 
PPTX
1 5 multiplication and division of rational expressions
bymath123b
 
PPTX
1 f7 on cross-multiplication
bymath123a
 
PPTX
14 the lcm and the multiplier method for addition and subtraction of rational...
byelem-alg-sample
 
PPTX
2 the least common multiple and clearing the denominators
bymath123b
 
PPTX
4 lcm and lcd 125s
byTzenma
 
PPTX
3 lcm and lcd, addition and subtraction of fractions
byelem-alg-sample
 
1 f4 lcm and lcd
bymath123a
 
14 lcm, addition and subtraction of fractions
byalg1testreview
 
1 5 multiplication and division of rational expressions
bymath123b
 
1 f7 on cross-multiplication
bymath123a
 
14 the lcm and the multiplier method for addition and subtraction of rational...
byelem-alg-sample
 
2 the least common multiple and clearing the denominators
bymath123b
 
4 lcm and lcd 125s
byTzenma
 
3 lcm and lcd, addition and subtraction of fractions
byelem-alg-sample
 

What's hot

PPTX
15 addition and subtraction of fractions
byalg-ready-review
 
PPTX
4 the lcm and clearing denominators x
byTzenma
 
PPTX
123a-1-f4 lcm and lcd
byNTIP
 
PPTX
1 f4 lcm and lcd
bymath123a
 
PPTX
Least common multiple (lcm) & greatest
byShiara Agosto
 
PPT
Common Multiples and Least Common Multiple
byBrooke Young
 
PDF
least common multiple
byAlejandra Elgueta
 
PPT
Ch. 5.1 - Least Common Multiple
byKathy Favazza
 
PPTX
Least common multiple
byMelanie_Anderson
 
PPT
3 6
bytriviamom
 
PPTX
Finds the common multiples and the least common demo teach
byrosalio baybayan jr
 
DOCX
Lesson 5 5 lcm
bymlabuski
 
PPT
Number theory.ppt22
byteena zacharias
 
PPTX
Common Multiples and Common Factors
byTim Bonnar
 
ODP
Questions on ratio and proportion
byDr. Trilok Kumar Jain
 
PPT
2nd9 interim review pp
byMs. Jones
 
PPT
Factors And Primes
byguesta8f50
 
15 addition and subtraction of fractions
byalg-ready-review
 
4 the lcm and clearing denominators x
byTzenma
 
123a-1-f4 lcm and lcd
byNTIP
 
1 f4 lcm and lcd
bymath123a
 
Least common multiple (lcm) & greatest
byShiara Agosto
 
Common Multiples and Least Common Multiple
byBrooke Young
 
least common multiple
byAlejandra Elgueta
 
Ch. 5.1 - Least Common Multiple
byKathy Favazza
 
Least common multiple
byMelanie_Anderson
 
3 6
bytriviamom
 
Finds the common multiples and the least common demo teach
byrosalio baybayan jr
 
Lesson 5 5 lcm
bymlabuski
 
Number theory.ppt22
byteena zacharias
 
Common Multiples and Common Factors
byTim Bonnar
 
Questions on ratio and proportion
byDr. Trilok Kumar Jain
 
2nd9 interim review pp
byMs. Jones
 
Factors And Primes
byguesta8f50
 

Viewers also liked

PPTX
English v
byBryan Mejia
 
DOC
Respuestas diapositiva 13 factorizacion de polinomios
byflor2510
 
PPTX
53 multiplication and division of rational expressions
byalg1testreview
 
PPTX
New Metrics for Marketing
byJennifer Edgerly
 
PDF
AAM CSTX 10 Hunter Ligger, CA
byDonald Moore
 
PPTX
peru
bymaria sanchez
 
PPTX
Gafas 3D
byHeydi94
 
PPTX
Ppt on karyotyping, chromosome banding and chromosome painting.
byICRISAT
 
PPTX
Aprendizaje autónomo-y-significativo
byIlse Paola Sánchez Ruiz
 
English v
byBryan Mejia
 
Respuestas diapositiva 13 factorizacion de polinomios
byflor2510
 
53 multiplication and division of rational expressions
byalg1testreview
 
New Metrics for Marketing
byJennifer Edgerly
 
AAM CSTX 10 Hunter Ligger, CA
byDonald Moore
 
peru
bymaria sanchez
 
Gafas 3D
byHeydi94
 
Ppt on karyotyping, chromosome banding and chromosome painting.
byICRISAT
 
Aprendizaje autónomo-y-significativo
byIlse Paola Sánchez Ruiz
 

Similar to 54 the least common multiple

PPTX
1 6 the least common multiple
bymath123b
 
PPTX
Finding the Least Common Multiple of Two or More Numbers
byalexbenitez64
 
PPT
Mathematics Lessonpowerpoint 8.13.2019.ppt
byBeccaGonzaga
 
PPT
Math lesson for junior high school for a
byQurrotuAyun20
 
PPT
Mathematics grade five quarter 1 Lesson.ppt
byBeccaGonzaga
 
PPT
Math Lesson 8.13sacCACdscszcsdc.2019.ppt
byBALQISNURAZIZAH1
 
PPT
Math Lesson the prime factorization method 8.13.2019.ppt
byAliAshraf291758
 
PPT
Ch. 5.1 Bt
byKathy Favazza
 
PPT
least common multiple.ppt
byCautES1
 
KEY
7-4 Common Multiples and LCM
byMel Anthony Pepito
 
KEY
7-4 Common Multiples and LCM
byRudy Alfonso
 
PPTX
fractions-least-common-multiple-1227659728817144-9.pptx
byPremkumarLetchumanan
 
PPTX
COT Q1. LCM.pptx
byLuisaGarcillan1
 
PPTX
Updated sec. 5.4
bybweldon
 
PPTX
Ciii. lesson-4-least-common-multiple
byErnestoAlfonso7
 
PPT
Least Common Multiple
byKathy Favazza
 
PPT
Least Common Multiple
byDonna Furrey
 
PPTX
Sec. 5.4
bybweldon
 
PDF
Factoring and multiples
byKenza Ismail
 
PPT
Gcf+lcm+word+problems powerpoint
byNeilfieOrit2
 
1 6 the least common multiple
bymath123b
 
Finding the Least Common Multiple of Two or More Numbers
byalexbenitez64
 
Mathematics Lessonpowerpoint 8.13.2019.ppt
byBeccaGonzaga
 
Math lesson for junior high school for a
byQurrotuAyun20
 
Mathematics grade five quarter 1 Lesson.ppt
byBeccaGonzaga
 
Math Lesson 8.13sacCACdscszcsdc.2019.ppt
byBALQISNURAZIZAH1
 
Math Lesson the prime factorization method 8.13.2019.ppt
byAliAshraf291758
 
Ch. 5.1 Bt
byKathy Favazza
 
least common multiple.ppt
byCautES1
 
7-4 Common Multiples and LCM
byMel Anthony Pepito
 
7-4 Common Multiples and LCM
byRudy Alfonso
 
fractions-least-common-multiple-1227659728817144-9.pptx
byPremkumarLetchumanan
 
COT Q1. LCM.pptx
byLuisaGarcillan1
 
Updated sec. 5.4
bybweldon
 
Ciii. lesson-4-least-common-multiple
byErnestoAlfonso7
 
Least Common Multiple
byKathy Favazza
 
Least Common Multiple
byDonna Furrey
 
Sec. 5.4
bybweldon
 
Factoring and multiples
byKenza Ismail
 
Gcf+lcm+word+problems powerpoint
byNeilfieOrit2
 

More from alg1testreview

PPTX
56 system of linear equations
byalg1testreview
 
PPTX
55 addition and subtraction of rational expressions
byalg1testreview
 
PPTX
52 rational expressions
byalg1testreview
 
PPTX
51 basic shapes and formulas
byalg1testreview
 
PPTX
41 expressions
byalg1testreview
 
PPTX
59 constructing linea equations of lines
byalg1testreview
 
PPTX
57 graphing lines from linear equations
byalg1testreview
 
PPTX
58 slopes of lines
byalg1testreview
 
PPTX
55 inequalities and comparative statements
byalg1testreview
 
PPTX
56 the rectangular coordinate system
byalg1testreview
 
PPTX
54 the number line
byalg1testreview
 
PPTX
53 pythagorean theorem and square roots
byalg1testreview
 
PPTX
52 about triangles
byalg1testreview
 
PPTX
50 solving equations by factoring
byalg1testreview
 
PPTX
51 ratio-proportion
byalg1testreview
 
PPTX
49 factoring trinomials the ac method and making lists
byalg1testreview
 
PPTX
48 factoring out the gcf and the grouping method
byalg1testreview
 
PPTX
47 operations of 2nd degree expressions and formulas
byalg1testreview
 
PPTX
45scientific notation
byalg1testreview
 
PPTX
46polynomial expressions
byalg1testreview
 
56 system of linear equations
byalg1testreview
 
55 addition and subtraction of rational expressions
byalg1testreview
 
52 rational expressions
byalg1testreview
 
51 basic shapes and formulas
byalg1testreview
 
41 expressions
byalg1testreview
 
59 constructing linea equations of lines
byalg1testreview
 
57 graphing lines from linear equations
byalg1testreview
 
58 slopes of lines
byalg1testreview
 
55 inequalities and comparative statements
byalg1testreview
 
56 the rectangular coordinate system
byalg1testreview
 
54 the number line
byalg1testreview
 
53 pythagorean theorem and square roots
byalg1testreview
 
52 about triangles
byalg1testreview
 
50 solving equations by factoring
byalg1testreview
 
51 ratio-proportion
byalg1testreview
 
49 factoring trinomials the ac method and making lists
byalg1testreview
 
48 factoring out the gcf and the grouping method
byalg1testreview
 
47 operations of 2nd degree expressions and formulas
byalg1testreview
 
45scientific notation
byalg1testreview
 
46polynomial expressions
byalg1testreview
 

Recently uploaded

PPTX
MEMORY &FORGETTING. Shilpa Hotakar.Psychology pptx
bySHILPA HOTAKAR
 
PPTX
WEEK 2 (2).pptx TLE COOKERY 10 QUARTER 4
byResynFayeCortez2
 
PPTX
Greengnorance Toolkit Module 3 Shopping and Food
byKarl Donert
 
PPTX
Plato's Insight model teaching and learning.pptx
byNikhitaDS
 
PPTX
Renal Physiology- Functional anatomy of Kidney.pptx
byDisha Soriya
 
PPTX
How to perform product search based on a custom field from the Website Shop p...
byCeline George
 
PDF
Chapter 05 Drugs Acting on the Central Nervous System: Anti-Depressant Drugs
bySandeshSul
 
PPTX
Greengnorance Toolkit Module 2 Energy saving
byKarl Donert
 
PPTX
Plant fibres used as surgical dressings & Sutures – Surgical Catgut and Ligat...
bySai Meer College of Pharmacy
 
PPTX
How to Add an Icon in Systray in Odoo 18
byCeline George
 
PDF
Pharmaceutical Quality Assurance Unit 1 (BP606T)
byChetan Mali
 
PDF
The Sheep and the Goat: Beckett’s Subversion of Divine Justice in Waiting for...
bysejadchokiya
 
PPTX
How to Create_Generate Engineering Change Orders ECOs in Odoo 18
byCeline George
 
PDF
Tetracycline Class of Antibiotics: SAR, MOA and Uses
bymominzarinamdnajeeb
 
PPTX
Greengnorance Toolkit Module 6 Water Resources
byKarl Donert
 
PDF
Orlando by Virginia Woolf: The Granite and The Rainbow
byAdityarajsinh Gohil
 
PPTX
Greengnorance Toolkit Module 5 Waste Management
byKarl Donert
 
PDF
BP502T Industrial Pharmacy I (Theory) UNIT– I (Part 1)
byRushi Mandali
 
PPTX
How to Manage Reservation Method in Odoo 18 Inventory
byCeline George
 
PDF
Family and Marriage __ Sociology __ Jhasketan Kuanar __ NURSING VIBE ONLY .pdf
byJK NURSING VIBES ONLY JHASKETAN KUANAR
 
MEMORY &FORGETTING. Shilpa Hotakar.Psychology pptx
bySHILPA HOTAKAR
 
WEEK 2 (2).pptx TLE COOKERY 10 QUARTER 4
byResynFayeCortez2
 
Greengnorance Toolkit Module 3 Shopping and Food
byKarl Donert
 
Plato's Insight model teaching and learning.pptx
byNikhitaDS
 
Renal Physiology- Functional anatomy of Kidney.pptx
byDisha Soriya
 
How to perform product search based on a custom field from the Website Shop p...
byCeline George
 
Chapter 05 Drugs Acting on the Central Nervous System: Anti-Depressant Drugs
bySandeshSul
 
Greengnorance Toolkit Module 2 Energy saving
byKarl Donert
 
Plant fibres used as surgical dressings & Sutures – Surgical Catgut and Ligat...
bySai Meer College of Pharmacy
 
How to Add an Icon in Systray in Odoo 18
byCeline George
 
Pharmaceutical Quality Assurance Unit 1 (BP606T)
byChetan Mali
 
The Sheep and the Goat: Beckett’s Subversion of Divine Justice in Waiting for...
bysejadchokiya
 
How to Create_Generate Engineering Change Orders ECOs in Odoo 18
byCeline George
 
Tetracycline Class of Antibiotics: SAR, MOA and Uses
bymominzarinamdnajeeb
 
Greengnorance Toolkit Module 6 Water Resources
byKarl Donert
 
Orlando by Virginia Woolf: The Granite and The Rainbow
byAdityarajsinh Gohil
 
Greengnorance Toolkit Module 5 Waste Management
byKarl Donert
 
BP502T Industrial Pharmacy I (Theory) UNIT– I (Part 1)
byRushi Mandali
 
How to Manage Reservation Method in Odoo 18 Inventory
byCeline George
 
Family and Marriage __ Sociology __ Jhasketan Kuanar __ NURSING VIBE ONLY .pdf
byJK NURSING VIBES ONLY JHASKETAN KUANAR
 

54 the least common multiple

  • 1.
    The Least CommonMultiple (LCM) Frank Ma © 2011
  • 2.
    The Least CommonMultiple (LCM) We say m is a multiple of x if m can be divided by x.
  • 3.
    For example, 12is a multiple of 2 The Least Common Multiple (LCM) We say m is a multiple of x if m can be divided by x.
  • 4.
    For example, 12is a multiple of 2. 12 is also a multiple of 3, 4, 6 or 12 itself. The Least Common Multiple (LCM) We say m is a multiple of x if m can be divided by x.
  • 5.
    For example, 12is a multiple of 2. 12 is also a multiple of 3, 4, 6 or 12 itself. The Least Common Multiple (LCM) xy2 is a multiple of x, We say m is a multiple of x if m can be divided by x.
  • 6.
    For example, 12is a multiple of 2. 12 is also a multiple of 3, 4, 6 or 12 itself. The Least Common Multiple (LCM) xy2 is a multiple of x, xy2 is also a multiple of y or xy. We say m is a multiple of x if m can be divided by x.
  • 7.
    For example, 12is a multiple of 2. 12 is also a multiple of 3, 4, 6 or 12 itself. The Least Common Multiple (LCM) A common multiple of two or more quantities is a multiple of all the given quantities. xy2 is a multiple of x, xy2 is also a multiple of y or xy. We say m is a multiple of x if m can be divided by x.
  • 8.
    For example, 12is a multiple of 2. 12 is also a multiple of 3, 4, 6 or 12 itself. The Least Common Multiple (LCM) 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 multiple of x, xy2 is also a multiple of y or xy. We say m is a multiple of x if m can be divided by x.
  • 9.
    For example, 12is a multiple of 2. 12 is also a multiple of 3, 4, 6 or 12 itself. The Least Common Multiple (LCM) 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 multiple of x, xy2 is also a multiple of y or xy. xy2 is a common multiple of x and y. We say m is a multiple of x if m can be divided by x.
  • 10.
    The least commonmultiple (LCM) of two or more numbers is the smallest common multiple of all the given numbers. For example, 12 is a multiple of 2. 12 is also a multiple of 3, 4, 6 or 12 itself. The Least Common Multiple (LCM) 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 multiple of x, xy2 is also a multiple of y or xy. xy2 is a common multiple of x and y. We say m is a multiple of x if m can be divided by x.
  • 11.
    The least commonmultiple (LCM) of two or more numbers is the smallest common multiple of all the given numbers. For example, 12 is a multiple of 2. 12 is also a multiple of 3, 4, 6 or 12 itself. For example, 12 is the LCM of 4 and 6. The Least Common Multiple (LCM) 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 multiple of x, xy2 is also a multiple of y or xy. xy2 is a common multiple of x and y. We say m is a multiple of x if m can be divided by x.
  • 12.
    The least commonmultiple (LCM) of two or more numbers is the smallest common multiple of all the given numbers. For example, 12 is a multiple of 2. 12 is also a multiple of 3, 4, 6 or 12 itself. For example, 12 is the LCM of 4 and 6, xy is the LCM of x and y, The Least Common Multiple (LCM) 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 multiple of x, xy2 is also a multiple of y or xy. xy2 is a common multiple of x and y. We say m is a multiple of x if m can be divided by x.
  • 13.
    The least commonmultiple (LCM) of two or more numbers is the smallest common multiple of all the given numbers. For example, 12 is a multiple of 2. 12 is also a multiple of 3, 4, 6 or 12 itself. 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. The Least Common Multiple (LCM) 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 multiple of x, xy2 is also a multiple of y or xy. xy2 is a common multiple of x and y. We say m is a multiple of x if m can be divided by x.
  • 14.
    The least commonmultiple (LCM) of two or more numbers is the smallest common multiple of all the given numbers. For example, 12 is a multiple of 2. 12 is also a multiple of 3, 4, 6 or 12 itself. 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. The Least Common Multiple (LCM) 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, We give two methods for finding the LCM. I. The searching method xy2 is a multiple of x, xy2 is also a multiple of y or xy. xy2 is a common multiple of x and y. We say m is a multiple of x if m can be divided by x.
  • 15.
    The least commonmultiple (LCM) of two or more numbers is the smallest common multiple of all the given numbers. For example, 12 is a multiple of 2. 12 is also a multiple of 3, 4, 6 or 12 itself. 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. The Least Common Multiple (LCM) 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, We give two methods for finding the LCM. I. The searching method II. The construction method xy2 is a multiple of x, xy2 is also a multiple of y or xy. xy2 is a common multiple of x and y. We say m is a multiple of x if m can be divided by x.
  • 16.
  • 17.
    Searching Method Given alist of quantities, list the multiples of the largest quantity and test them one by one until we find the LCM. Methods of Finding LCM
  • 18.
    Searching Method Given alist 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. Methods of Finding LCM
  • 19.
    Searching Method Given alist 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: Methods of Finding LCM
  • 20.
    Searching Method Given alist 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: 24, 48, 72, Methods of Finding LCM
  • 21.
    Searching Method Given alist 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: 24, 48, 72, 96, 120, 144, … Methods of Finding LCM
  • 22.
    Searching Method Given alist 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: 24, 48, 72, 96, 120, 144, … so the LCM of {18, 24, 16} is 144. Methods of Finding LCM
  • 23.
    Searching Method Given alist 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: 24, 48, 72, 96, 120, 144, … so the LCM of {18, 24, 16} is 144. Methods of Finding LCM Construction Method:
  • 24.
    Searching Method Given alist 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: 24, 48, 72, 96, 120, 144, … so the LCM of {18, 24, 16} is 144. Methods of Finding LCM Construction Method: Given two or more quantities, I. factor each quantity completely,
  • 25.
    Searching Method Given alist 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: 24, 48, 72, 96, 120, 144, … so the LCM of {18, 24, 16} is 144. Methods of Finding LCM 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.
  • 26.
    Searching Method Given alist 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: 24, 48, 72, 96, 120, 144, … so the LCM of {18, 24, 16} is 144. Methods of Finding LCM Example B. a. Construct the LCM of 18, 24, 20. 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.
  • 27.
    Searching Method Given alist 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: 24, 48, 72, 96, 120, 144, … so the LCM of {18, 24, 16} is 144. Methods of Finding LCM Example B. a. Construct the LCM of 18, 24, 20. Factor each number 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.
  • 28.
    Searching Method Given alist 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: 24, 48, 72, 96, 120, 144, … so the LCM of {18, 24, 16} is 144. Methods of Finding LCM Example B. a. Construct the LCM of 18, 24, 20. Factor each number completely 18 = 2*32 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.
  • 29.
    Searching Method Given alist 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: 24, 48, 72, 96, 120, 144, … so the LCM of {18, 24, 16} is 144. Methods of Finding LCM Example B. a. Construct the LCM of 18, 24, 20. Factor each number completely 18 = 2*32 24 = 233 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.
  • 30.
    Searching Method Given alist 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: 24, 48, 72, 96, 120, 144, … so the LCM of {18, 24, 16} is 144. Methods of Finding LCM Example B. a. Construct the LCM of 18, 24, 20. Factor each number completely 18 = 2*32 24 = 233 20 = 225 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.
  • 31.
    Searching Method Given alist 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: 24, 48, 72, 96, 120, 144, … so the LCM of {18, 24, 16} is 144. Methods of Finding LCM Example B. a. Construct the LCM of 18, 24, 20. Factor each number completely 18 = 2*32 24 = 233 20 = 225 Take the highest of the each factor, 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.
  • 32.
    Searching Method Given alist 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: 24, 48, 72, 96, 120, 144, … so the LCM of {18, 24, 16} is 144. Methods of Finding LCM Example B. a. Construct the LCM of 18, 24, 20. Factor each number completely 18 = 2*32 24 = 233 20 = 225 Take the highest of the each factor, 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.
  • 33.
    Searching Method Given alist 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: 24, 48, 72, 96, 120, 144, … so the LCM of {18, 24, 16} is 144. Methods of Finding LCM Example B. a. Construct the LCM of 18, 24, 20. Factor each number completely 18 = 2*32 24 = 233 20 = 225 Take the highest of the each factor, 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.
  • 34.
    Searching Method Given alist 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: 24, 48, 72, 96, 120, 144, … so the LCM of {18, 24, 16} is 144. Methods of Finding LCM Example B. a. Construct the LCM of 18, 24, 20. Factor each number completely 18 = 2*32 24 = 233 20 = 225 Take the highest of the each factor, 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.
  • 35.
    Searching Method Given alist 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: 24, 48, 72, 96, 120, 144, … so the LCM of {18, 24, 16} is 144. Methods of Finding LCM Example B. a. Construct the LCM of 18, 24, 20. Factor each number completely 18 = 2*32 24 = 233 20 = 225 Take the highest of the each factor, so the LCM is 23325 = 360. 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.
  • 36.
    b. Construct theLCM of x2y3z, x3yz4, x3zw Methods of Finding LCM
  • 37.
    b. Construct theLCM of x2y3z, x3yz4, x3zw All quantities are factored, Take the highest power of each factor. Methods of Finding LCM
  • 38.
    b. Construct theLCM of x2y3z, x3yz4, x3zw All quantities are factored, Take the highest power of each factor. The LCM = Methods of Finding LCM
  • 39.
    b. Construct theLCM of x2y3z, x3yz4, x3zw All quantities are factored, Take the highest power of each factor. The LCM = x3 Methods of Finding LCM
  • 40.
    b. Construct theLCM of x2y3z, x3yz4, x3zw All quantities are factored, Take the highest power of each factor. The LCM = x3y3 Methods of Finding LCM
  • 41.
    b. Construct theLCM of x2y3z, x3yz4, x3zw All quantities are factored, Take the highest power of each factor. The LCM = x3y3z4 Methods of Finding LCM
  • 42.
    b. Construct theLCM of x2y3z, x3yz4, x3zw All quantities are factored, Take the highest power of each factor. The LCM = x3y3z4w. Methods of Finding LCM
  • 43.
    b. Construct theLCM of x2y3z, x3yz4, x3zw All quantities are factored, Take the highest power of each factor. The LCM = x3y3z4w. Methods of Finding LCM c. Construct the LCM of x2 – 3x + 2, x2 – x – 2
  • 44.
    b. Construct theLCM of x2y3z, x3yz4, x3zw All quantities are factored, Take the highest power of each factor. The LCM = x3y3z4w. Methods of Finding LCM c. Construct the LCM of x2 – 3x + 2, x2 – x – 2 Factor each quantity.
  • 45.
    b. Construct theLCM of x2y3z, x3yz4, x3zw All quantities are factored, Take the highest power of each factor. The LCM = x3y3z4w. Methods of Finding LCM c. Construct the LCM of x2 – 3x + 2, x2 + x – 2 Factor each quantity. x2 – 3x + 2 = (x – 2)(x – 1)
  • 46.
    b. Construct theLCM of x2y3z, x3yz4, x3zw All quantities are factored, Take the highest power of each factor. The LCM = x3y3z4w. Methods of Finding LCM c. Construct the LCM of x2 – 3x + 2, x2 + x – 2 Factor each quantity. x2 – 3x + 2 = (x – 2)(x – 1) x2 + x – 2 = (x + 2)(x – 1)
  • 47.
    b. Construct theLCM of x2y3z, x3yz4, x3zw All quantities are factored, Take the highest power of each factor. The LCM = x3y3z4w. Methods of Finding LCM c. Construct the LCM of x2 – 3x + 2, x2 + 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 =
  • 48.
    b. Construct theLCM of x2y3z, x3yz4, x3zw All quantities are factored, Take the highest power of each factor. The LCM = x3y3z4w. Methods of Finding LCM c. Construct the LCM of x2 – 3x + 2, x2 + 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)
  • 49.
    b. Construct theLCM of x2y3z, x3yz4, x3zw All quantities are factored, Take the highest power of each factor. The LCM = x3y3z4w. Methods of Finding LCM c. Construct the LCM of x2 – 3x + 2, x2 + 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)
  • 50.
    b. Construct theLCM of x2y3z, x3yz4, x3zw All quantities are factored, Take the highest power of each factor. The LCM = x3y3z4w. Methods of Finding LCM c. Construct the LCM of x2 – 3x + 2, x2 + 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)(x +2)
  • 51.
    b. Construct theLCM of x2y3z, x3yz4, x3zw All quantities are factored, Take the highest power of each factor. The LCM = x3y3z4w. Methods of Finding LCM d. Construct the LCM of x4 – 4x2, x2 – 4x +4 c. Construct the LCM of x2 – 3x + 2, x2 + 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)(x +2)
  • 52.
    b. Construct theLCM of x2y3z, x3yz4, x3zw All quantities are factored, Take the highest power of each factor. The LCM = x3y3z4w. Methods of Finding LCM d. Construct the LCM of x4 – 4x2, x2 – 4x +4 Factor each quantity. c. Construct the LCM of x2 – 3x + 2, x2 + 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)(x +2)
  • 53.
    b. Construct theLCM of x2y3z, x3yz4, x3zw All quantities are factored, Take the highest power of each factor. The LCM = x3y3z4w. Methods of Finding LCM d. Construct the LCM of x4 – 4x2, x2 – 4x +4 Factor each quantity. x4 – 4x2 = x2(x2 – 4) c. Construct the LCM of x2 – 3x + 2, x2 + 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)(x +2)
  • 54.
    b. Construct theLCM of x2y3z, x3yz4, x3zw All quantities are factored, Take the highest power of each factor. The LCM = x3y3z4w. Methods of Finding LCM d. Construct the LCM of x4 – 4x2, x2 – 4x +4 Factor each quantity. x4 – 4x2 = x2(x2 – 4) = x2(x + 2)(x – 2) c. Construct the LCM of x2 – 3x + 2, x2 + 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)(x +2)
  • 55.
    b. Construct theLCM of x2y3z, x3yz4, x3zw All quantities are factored, Take the highest power of each factor. The LCM = x3y3z4w. Methods of Finding LCM d. Construct the LCM of x4 – 4x2, x2 – 4x +4 Factor each quantity. x4 – 4x2 = x2(x2 – 4) = x2(x + 2)(x – 2) x2 – 4x + 4 = (x – 2)2 c. Construct the LCM of x2 – 3x + 2, x2 + 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)(x +2)
  • 56.
    b. Construct theLCM of x2y3z, x3yz4, x3zw All quantities are factored, Take the highest power of each factor. The LCM = x3y3z4w. Methods of Finding LCM d. Construct the LCM of x4 – 4x2, x2 – 4x +4 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 = c. Construct the LCM of x2 – 3x + 2, x2 + 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)(x +2)
  • 57.
    b. Construct theLCM of x2y3z, x3yz4, x3zw All quantities are factored, Take the highest power of each factor. The LCM = x3y3z4w. Methods of Finding LCM d. Construct the LCM of x4 – 4x2, x2 – 4x +4 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 c. Construct the LCM of x2 – 3x + 2, x2 + 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)(x +2)
  • 58.
    b. Construct theLCM of x2y3z, x3yz4, x3zw All quantities are factored, Take the highest power of each factor. The LCM = x3y3z4w. Methods of Finding LCM d. Construct the LCM of x4 – 4x2, x2 – 4x +4 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 c. Construct the LCM of x2 – 3x + 2, x2 + 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)(x +2)
  • 59.
    b. Construct theLCM of x2y3z, x3yz4, x3zw All quantities are factored, Take the highest power of each factor. The LCM = x3y3z4w. Methods of Finding LCM d. Construct the LCM of x4 – 4x2, x2 – 4x +4 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) c. Construct the LCM of x2 – 3x + 2, x2 + 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)(x +2)
  • 60.
    The Construction Methodtakes just enough of each factor to make the LCM which "covers" all the quantities on the list. Methods of Finding LCM
  • 61.
    The Construction Methodtakes 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 LCM
  • 62.
    Methods of FindingLCM 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?
  • 63.
    Science Math. EnglishArts A 2 yrs 3 yrs 3 yrs B 3 yrs 3 yrs 2 yrs C 2 yrs 2 yrs 4 yrs 1 yrs Methods of Finding LCM 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?
  • 64.
    Science Math. EnglishArts A 2 yrs 3 yrs 3 yrs B 3 yrs 3 yrs 2 yrs C 2 yrs 2 yrs 4 yrs 1 yrs Methods of Finding LCM 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?
  • 65.
    Science Math. EnglishArts A 2 yrs 3 yrs 3 yrs B 3 yrs 3 yrs 2 yrs C 2 yrs 2 yrs 4 yrs 1 yrs Science-3 yr Math-3yrs English-4 yrs Arts-1yr Methods of Finding LCM 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?
  • 66.
    Science Math. EnglishArts A 2 yrs 3 yrs 3 yrs B 3 yrs 3 yrs 2 yrs C 2 yrs 2 yrs 4 yrs 1 yrs Science-3 yr Math-3yrs English-4 yrs Arts-1yr Methods of Finding LCM 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? So we take the highest power of each factor to make the LCM.
  • 67.
    Ex. A. Findthe LCM of the given expressions. Methods of Finding LCM xy3, xy1. x2y, xy32. xy3, xyz3. 4xy3, 6xy4. x2y, xy3, xy25. 9xy3, 12x3y, 15x2y36. (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)x18. 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 – 421. x2 – x – 6, x2 – x – 222. x2 + 2x – 3, 1 – x225. –x2 – 2x + 3, x2 – x26. x2 – x – 6, x2 – 923. x2 + 5x – 6, x2 – x – 624. 2x2 + 5x – 3, x – 4x327. x2 – 5x + 4, x3 – 16x28.