Transcript prime factorization

Welcome to math!
Thursday, July 28th, 2011
1. Turn in your syllabus and e-mail verification form
on the front table if you have not already done so.
2. Warm Up – write in your notes for today
Write each number as a product of two whole numbers in as
many ways as possible (write the fact families).
6 1 x 6, 2 x 3
16 1 x 16, 2 x 8, 4 x 4
17 1 x 17
36 1 x 36, 2 x 18, 3 x 12, 4 x 9, 6 x 6
5. 23 1 x 23
1.
2.
3.
4.
3. Objective: Find the prime factorizations of composite numbers.
(write in notes)
4. Definitions: prime number, composite number and prime factorization.
5. Grab a whiteboard if you have your own marker and eraser.
6. Work on your MyFace Activity when finished.
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Vocabulary
prime number
composite number
prime factorization
A prime number is a whole number greater
than 1 that has exactly two factors, 1 and itself.
Three is a prime number because its only
factors are 1 and 3.
A composite number is a whole number that
has more than two factors. Six is a composite
number because it has more than two
factors—1, 2, 3, and 6. The number 1 has
exactly one factor and is neither prime
nor composite.
A composite number can be written as the
product of its prime factors. This is called the
prime factorization of the number.
You can use a factor tree to find the prime
factors of a composite number.
Additional Example 2A & 2B: Identifying Prime and
Composite Numbers
Tell whether each number is prime or
composite.
A. 23
divisible by 1, 23
prime
B. 48
divisible by 1, 2, 3, 4, 6, 8, 12, 16, 24, 48.
composite
Additional Example 2C & 2D: Identifying Prime and
Composite Numbers
Tell whether each number is prime or
composite.
C. 31
divisible by 1, 31
prime
D. 18
divisible by 1, 2, 3, 6, 9, 18
composite
Try This: Example 2A & 2B
Tell whether each number is prime or
composite.
A. 27
divisible by 1, 3, 9, 27
composite
B. 24
divisible by 1, 2, 3, 4, 6, 8, 12, 24
composite
Divisibility Rules
A number is divisible by. . .
Divisible
Not Divisible
2
if the last digit is even (0, 2, 4, 6, or 8).
3,978
4,975
3
if the sum of the digits is divisible by 3.
315
139
4
if the last two digits form a number
divisible by 4.
8,512
7,518
5
if the last digit is 0 or 5.
14,975
10,978
6
if the number is divisible by both 2 and 3
48
20
9
if the sum of the digits is divisible by 9.
711
93
15,990
10,536
10 if the last digit is 0.
Example 1: Using a Factor Tree to Find Prime
Factorization
Write the prime factorization of the number.
A. 24
24
8 · 3
4 · 2 · 3
Write 24 as the product of
two factors.
Continue factoring until all
factors are prime.
2 · 2 · 2 · 3
The prime factorization of 24 is 2 · 2 · 2 · 3. Using
exponents, you can write this as 23 · 3.
Example B: Using a Factor Tree to Find Prime
Factorization
Write the prime factorization of the number.
B. 150
150
30 · 5
10 · 3 · 5
2 · 5 · 3 · 5
Write 150 as the product
of two factors.
Continue factoring until
all factors are prime.
The prime factorization of 150 is 2 · 3 · 5 · 5, or
2 · 3 · 52.
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Try This: Example 1A
Write the prime factorization of the number.
A. 36
36
18 · 2
9 · 2 · 2
Write 36 as the product of
two factors.
Continue factoring until all
factors are prime.
3 · 3 · 2 · 2
The prime factorization of 36 is 2 · 2 · 3 · 3. Using
exponents, you can write this as 22 · 32.
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Try This: Example 1B
Write the prime factorization of the number.
B. 90
90
45 · 2
9 · 5 · 2
3 · 3 · 5 · 2
Write 90 as the product
of two factors.
Continue factoring until
all factors are prime.
The prime factorization of 90 is 3 · 3 · 5 · 2, or
2 · 32 · 5.
You can also use a step diagram to find the prime
factorization of a number…
Steps for Using the Step Diagram for Prime
Factorization
1. At each step, divide by the smallest possible
prime number.
2. Continue dividing until the quotient is 1.
The prime factors of the number are the prime
numbers you divided by.
Example 3: Using a Step Diagram to Find Prime
Factorization
Write the prime factorization of each number.
A. 476
2 476
2 238
7 119
17 17
1
Divide 476 by 2. Write the
quotient below 476.
Keep dividing by a prime
number.
Stop when the quotient is 1.
The prime factorization of 476 is 2 · 2 · 7 · 17, or
22 · 7 · 17.
Additional Example 2B: Using a Step Diagram to Find
Prime Factorization
Write the prime factorization of the number.
B. 275
5 275
5 55
11 11
1
Divide 275 by 5. Write the quotient
below 275.
Stop when the quotient is 1.
The prime factorization of 275 is 5 · 5 · 11, or
52 · 11.
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Try This: Example 4
Write the prime factorization of each number.
A. 324
2 324
2 162
3 81
3 27
3 9
3 3
1
The prime factorization
22 · 34.
Divide 324 by 2. Write the
quotient below 324.
Keep dividing by a prime
number.
Stop when the quotient is 1.
of 324 is 2 · 2 · 3 · 3 · 3 · 3, or
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Try This: Example 2B
Write the prime factorization of the number.
B. 325
5 325
5 65
13 13
1
Divide 325 by 5. Write the quotient
below 325.
Stop when the quotient is 1.
The prime factorization of 325 is 5 · 5 · 13, or
52 · 13.
There is only one prime factorization for any
given composite number. Example 2A began by
dividing 476 by 2, the smallest prime factor of
476. Beginning with any prime factor of 476
gives the same result.
2 476
2 238
7 119
17 17
1
7 476
2 68
2 34
17 17
1
The prime factorizations are 2 · 2 · 7 · 17 and
7 · 2 · 2 · 17, which are the same as 17 · 2 · 2 · 7.