Transcript Factors and Prime Factorization

Do Now 1/4/10
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Copy HW in your planner.
– Text p. 214, #1-17 all (you need your text)
– Chapter 4 Test Wednesday
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Be ready to review sections 4.1 – 4.4.
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Be ready to copy POTW #6.
Objective
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SWBAT review prime factorization,
greatest common factor, equivalent
fractions, and least common multiple
Section 4.1 “Factors and Prime
Factorization”
Natural numbers are classified according to how
many factors they have:
(1) Prime numbers:
-a whole number that is greater than 1 and has exactly
TWO factors, itself and 1.
(2) Composite numbers:
-a whole number that is greater than 1 and has
more than two factors.
(3) The number 1:
1 is neither prime nor composite.
Factoring Natural Numbers
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You can use TREE DIAGRAMS to factor
a number until all factors are primes.
30
30
30
3 ∙ 10
2 ∙ 15
5 ∙ 6
3 ∙ 2 ∙ 5
2 ∙ 3 ∙ 5
5 ∙ 2 ∙ 3
**HINT
Write your factors
in increasing order.
Prime Factorization
-writing a number as a product of prime factors.
Monomial
a number, a variable, or the product of a
number and one or more variables with whole
number exponents
7
x
3x³yz²
To FACTOR a monomial, write the monomial as a
product of prime numbers and variables with
exponents of 1.
3a³ = 3 · a · a · a
Section 4.2 “Greatest Common Factor”
Greatest Common Factor
The LARGEST common factor of two numbers.
Find the GCF of 8 and 12.
List the factors of the two numbers and then compare.
8: 1, 2, 4, 8
12: 1, 2, 3, 4, 6, 12
From the list you can see that 1, 2, and 4 are common factors.
Of these 4 is the greatest common factor.
Greatest Common Factors
Find the GCF of 39 and 50.
When the numbers are too large to list the factors
of the two numbers, find the prime factorization of each
and then compare.
39: 3 · 13
50: 2 · 5 · 5
Relatively Prime:
Two numbers are relatively
prime if their greatest common
factor is 1.
From the list you can see that there are NO common prime
factors. However, two numbers always have 1 as a common
factor. So the GCF is 1 and the two numbers are relatively
prime.
GCF of Variable Expressions
Find the GCF of 18xy² and 28x²y³.
Write the prime factorization of each and then compare.
18xy²:
28x²y³:
2·3·3·x·y·y
2·2·7·x·x·y·y·y
From the list you can see that 2, x, y, and y are common prime
factors. The greatest common factor of is the
product of 2 · x · y · y, which is 2xy².
Section 4.3 “Equivalent Fractions”
EQUIVALENT FRACTIONStwo fractions are equivalent if they
represent the same quantity.
1
4
=
2
8
2
5
=
10
25
A fraction is in SIMPLEST FORM if its
numerator and denominator have a greatest
common factor of 1.
Simplifying Fractions
When simplifying fractions, look for the greatest
common factor in the numerator and denominator.
Then divide.
2
12
26
=
=
3
18
36
4
28
47
=
=
9
63 9  7
Simplify Variable Expressions
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To simplify a variable expression, you can
factor the numerator and denominator and
then divide out any common factors.
4 3 x
3
12 x


2
3
4 x x x
x
4x
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A variable expression is in SIMPLEST
FORM if the numerator and denominator
have no factors in common other than 1.
Section 4.4 “Least Common Multiple”
A multiple of a number is the
product of the number and any other
number greater than zero.
What are the multiples of 5?
Think of counting by fives…
5: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55 …
Least Common Multiple
Of all the common multiples of two
numbers, the smallest is the least common
multiple.
Least Common Multiple
Find the LCM of 64 and 72.
When the numbers are too large to list the multiples
of the two numbers, find the prime factorization of each
number. The LCM is the product of the common prime
factors and the factors that are not common.
64: 2 · 2 · 2 · 2 · 2 · 2
Common
Factors
2·2·2
72: 2 · 2 · 2 · 3 · 3
Not Common
Factors
2·2·2
3·3
From the list you can see that 2, 2, and 2 are common prime
factors, and the factors that are not common are 2, 2, 2, 3, and 3.
The LCM is then 2 · 2 · 2 · 2 · 2 · 2 · 3 · 3 which is 576.
Least Common Multiple
Find the LCM of 2a³b and 3ab.5
The LCM is the product of the common prime
factors and the factors that are not common.
Common Factors
2a³b : 2 · a · a · a · b
5
3ab : 3 · a · b · b · b · b · b
a·b
Not Common Factors
2·a·a
3·b·b·b·b
From the list you can see that a and b are common prime
factors, and the factors that are not common are 2, 3, a, a, b,
b, b, and b. The LCM is then a · b · 2 · a · a · 3 · b · b · b · b
5
which is 6a³b.
B-I-N-G-O
Complete numbers #6, 8, 10, 12 –
32 all on page 806.
Using your correct answers we will
play BINGO.
BINGO Review
Text p. 806, #6, 8, 10, 12-32 all
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6) prime
8) 2  2  2  2  5  5
10) 1; yes
12) 8; no
13) 4y
14) 4p
15) 2s
16) 3x
17) 3/16
18) 2/21
19) 7/11
20) 2/5
21) 2
3m
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22)
6
rs
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23)
19 ab
2c
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24)
25 x 3
6
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30
84
480
90
90c 2
36s 3
80n 2 p
741z 4
Homework
Text p. 214, #1-17 all