Download 4) Prime Factorization: every composite number can be expressed

Section3.1_Factors_and_Multiples_of_Whole_Numbers.notebook
October 14, 2014
Unit 3: Roots and Powers
Section 3.1: Factors and Multiples of Whole Numbers
Definitions
1) A factor is a number that divides exactly into another number.
example: 3 and 7 are factors of 21 because 3 x 7 = 21
2) Prime Number: a whole number with exactly two factors, 1 and itself.
examples:
*Note:
2, 3, 19, etc
0 and 1 are NOT prime numbers
3) Composite Number: a number with more than 2 factors.
example:
8 (factors: 1, 2, 4, 8)
Example:
Sort these numbers.
Natural Numbers
Prime Numbers
Composite Numbers
Nov 27­12:30 PM
4) Prime Factorization: every composite number can be expressed as a product of prime factors
Examples
1.
24 2.
500 Nov 27­2:18 PM
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Section3.1_Factors_and_Multiples_of_Whole_Numbers.notebook
October 14, 2014
Example: Write the prime factorization of the following.
A) 54 B) 512
Method 1: Factor Tree
Method 2: Repeated division by prime factors. Keep dividing by prime factors until your quotient is 1.
Nov 25­7:58 AM
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#'s 4 & 6
Nov 25­7:59 AM
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Section3.1_Factors_and_Multiples_of_Whole_Numbers.notebook
October 14, 2014
Greatest Common Factor (GCF): The greatest number that divides into each number in a set.
Example:
Find the GCF of 12 and 18.
Method 1:
List the factors of each number and chose the largest common factor.
Method 2:
Prime Factorization (Product of Primes)
The GCF is the product of the Minimum Power of each prime factor.
12 = 2 x 2 x 3
= 22 x 3 18 = 2 x 3 x 3 = 2 x 32
Therefore, the GCF is 2 x 3 = 6.
Nov 27­3:22 PM
Example:
Determine the GCF of:
A) 90, 54, and 72.
B) 75, 200, and 250
Nov 27­3:57 PM
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Section3.1_Factors_and_Multiples_of_Whole_Numbers.notebook
October 14, 2014
Example: Reduce the following fractions by finding the CGF.
B)
A)
Nov 26­8:32 AM
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#'s 4 ­­ 6, 8, 9, 15 Nov 26­8:10 AM
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Section3.1_Factors_and_Multiples_of_Whole_Numbers.notebook
October 14, 2014
Least Common Multiple (LCM): of two or more numbers is the smallest number (multiple) that is divisible by each number.
Example: Consider the numbers 6 and 8.
6x8 = 48 So, 48 is a multiple of 6 and 8...but it is not the LCM.
How do we find the LCM? 1) List the multiples of each number, unitl the same number appears in each list. That number is the LCM.
Multiples of 6:
Multiples of 8:
6, 12, 18, 24, 30, 36 etc.
8, 16, 24, 32, 40, etc.
The LCM is 24.
2) Prime Factorization (or Product of Primes)
6 = 2 x 3
8 = 4 x 2 = 2 x 2 x 2
= 23
The LCM is the product of the Maximum Power of each prime factor. So LCM = 23 x 3 = 8 x 3 = 24
Nov 27­4:03 PM
Example:
Find the LCM of the following. Use two different methods.
A) 27 and 45 B) 28, 42 and 63
Nov 27­4:24 PM
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Section3.1_Factors_and_Multiples_of_Whole_Numbers.notebook
Example:
October 14, 2014
Pencils come in packages of 10. Erasers come in packages of 12. Jason wants to purchase the smallest number of pencils and erasers so that he will have exactly 1 eraser per pencil. How many packages of pencils and erasers should Jason buy?
Dec 3­8:10 AM
Example:
One trip around a track is 440 yards. One runner can complete one lap in 8 minutes, the other runner can complete it in 6 minutes. How long will it take for both runners to arrive at their starting point together if they start at the same time and maintain their pace? Dec 3­8:10 AM
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Section3.1_Factors_and_Multiples_of_Whole_Numbers.notebook
Example:
October 14, 2014
What is the side length of the smallest square that
could be tiled with rectangles that measure 8 in. by 36 in.?Assume the rectangles cannot be cut. Sketch the square and rectangles.
Nov 27­4:44 PM
Example:
What is the side length of the largest square that could be used to tile a rectangle that measures 8 in. by 36 in.? Assume that the squares cannot be cut. Sketch the rectangle and squares.
Nov 25­7:55 AM
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