Download 4-2 Factors and Prime Factorization

4-2 Factors and Prime Factorization
Warm Up
Identify each number as prime or
composite.
1. 19
prime
2. 82
composite
3. 57
composite
4. 85
composite
5. 101
prime
6. 121 composite
Course 1
Wednesday, September 14, 11
4-2 Factors and Prime Factorization
Learn to write prime factorizations of
composite numbers.
Course 1
Wednesday, September 14, 11
4-2 Factors
Insert Lesson
TitleFactorization
Here
and Prime
Vocabulary
factor
prime factorization
Fundamental Theorem of
Arithmetic
exponent
Course 1
Wednesday, September 14, 11
4-2 Factors and Prime Factorization
Whole numbers that are multiplied to
find a product are called factors of that
product. A number is divisible by its
factors.
2 3=6
Factors
Product
Course 1
Wednesday, September 14, 11
6 ÷3 = 2
6 ÷2 = 3
6 is divisible
by 3 and 2.
4-2 Factors and Prime Factorization
Helpful Hint
When the pairs of factors begin to repeat,
then you have found all of the factors of
the number you are factoring.
Course 1
Wednesday, September 14, 11
4-2 Factors and Prime Factorization
Additional Example 1A: Finding Factors
List all factors of 16.
Begin listing factors in pairs.
16 = 1 • 16
16 = 2 • 8
16 = 4 • 4
16 = 8 • 2
1 2 4
4
1
2
3
4
5
6
7
8
is a factor.
is a factor.
is not a factor.
is a factor.
is not a factor.
is not a factor.
is not a factor.
and 2 have already been listed so stop here.
8 16
You can draw a diagram to illustrate the
factor pairs.
The factors of 16 are 1, 2, 4, 8, and 16.
Course 1
Wednesday, September 14, 11
4-2 Factors and Prime Factorization
Additional Example 1B: Finding Factors
List all factors of 19.
Begin listing all factors in pairs.
19 = 1 • 19
19 is not divisible by any
other whole number.
The factors of 19 are 1 and 19.
Course 1
Wednesday, September 14, 11
4-2 Factors and Prime Factorization
Check It Out: Example 1A
List all factors of 12.
Begin listing factors in pairs.
12
12
12
12
=
=
=
=
1
2
3
4
•
•
•
•
1 2 3
12
6
4
3
1 is a factor.
2 is a factor.
3 is a factor.
4 and 3 have already been listed
so stop here.
4 6 12
You can draw a diagram to
illustrate the factor pairs.
The factors of 12 are 1, 2, 3, 4, 6, and 12
Course 1
Wednesday, September 14, 11
4-2 Factors and Prime Factorization
Check It Out: Example 1B
List all factors of 11.
Begin listing all factors in pairs.
11 = 1 • 11
11 is not divisible by any
other whole number.
The factors of 11 are 1 and 11.
Course 1
Wednesday, September 14, 11
4-2 Factors and Prime Factorization
You can use factors to write a number
in different ways.
Factorization of 12
1 • 12
2•6
3•4
3•2•2
Notice that
these factors
are all prime.
The prime factorization of a number is
the number written as the product of its
prime factors. This is also called the
Fundamental Theorom of Arithmetic.
Course 1
Wednesday, September 14, 11
4-2 Factors and Prime Factorization
Helpful Hint
You can use exponents to write prime
factorizations. Remember that an
exponent tells you how many times the
base is a factor.
5
3 = 3x3x3x3x3
Course 1
Wednesday, September 14, 11
4-2 Factors and Prime Factorization
Additional Example 2A: Writing Prime Factorizations
Write the prime factorization of 24.
Method 1: Use a factor tree.
Choose any two factors of 24 to begin. Keep finding
factors until each branch ends at a prime factor.
24
2
•
2
24
•
6
12
•
6
2
•
3
3
•
2
4
2
•
2
24 = 3 • 2 • 2 • 2
24 = 2 • 2 • 2 • 3
The prime factorization of 24 is 2 • 2 • 2 • 3, or 23 • 3.
Course 1
Wednesday, September 14, 11
4-2 Factors and Prime Factorization
Additional Example 2B: Writing Prime Factorizations
Write the prime factorization of 45.
Method 2: Use a ladder diagram.
Choose a prime factor of 45 to begin. Keep dividing by
prime factors until the quotient is 1.
3
45
5
15
3
5
5
1
45 = 3 • 3 • 5
45
9
3
3
3
1
45 = 5 • 3 • 3
The prime factorization of 45 is 3 • 3 • 5 or 32 • 5 .
Course 1
Wednesday, September 14, 11
4-2 Factors and Prime Factorization
In Example 2, notice that the prime
factors may be written in a different order,
but they are still the same factors. Except
for changes in the order, there is only one
way to write the prime factorization of a
number.
Course 1
Wednesday, September 14, 11
4-2 Factors and Prime Factorization
Check It Out: Example 2A
Write the prime factorization of 28.
Method 1: Use a factor tree.
Choose any two factors of 28 to begin. Keep finding
factors until each branch ends at a prime factor.
28
2
•
2
28
14
•
7
7
28 = 2 • 2 • 7
•
4
2
•
2
28 = 7 • 2 • 2
The prime factorization of 28 is 2 • 2 • 7, or 22 • 7 .
Course 1
Wednesday, September 14, 11
4-2 Factors and Prime Factorization
Check It Out: Example 2B
Write the prime factorization of 36.
Method 2: Use a ladder diagram.
Choose a prime factor of 36 to begin. Keep
dividing by prime factors until the quotient is 1.
3
36
3
12
2
3
12
3
6
2
36
4
2
3
1
36 = 3 • 2 • 2 • 3
2
2
1
36 = 3 • 3 • 2 • 2
The prime factorization of 36 is 3 • 2 • 2 • 3, or 32 • 22.
Course 1
Wednesday, September 14, 11
4-2 Factors
Insert Lesson
and Prime
TitleFactorization
Here
Lesson Quiz
List all the factors of each number.
1. 22
1, 2, 11, 22
2. 40
1, 2, 4, 5, 8, 10, 20, 40
3. 51
1, 3, 17, 51
Write the prime factorization of each number.
5
2
4. 32
5. 120
Course 1
Wednesday, September 14, 11
23 × 3 × 5