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3-1 Prime Factorization
Warm Up
Write each number as a product of two whole
numbers in as many ways as possible.
1. 6
1 · 6, 2 · 3
2. 16
1 · 16, 2 · 8, 4 · 4
3. 17
1 · 17
4. 36
1 · 36, 2 · 18, 3 · 12, 4 · 9, 6 · 6
5. 23
1 · 23
Course 2
3-1 Prime Factorization
Learn to find the prime factorizations of
composite numbers.
Course 2
3-1 Prime
Insert Factorization
Lesson Title Here
Vocabulary
prime number
composite number
prime factorization
Course 2
3-1 Prime Factorization
In June 1999, Nayan Hajratwala discovered the
first known prime number with more than one
million digits. The new prime number, 26,972,593 – 1,
has 2,098,960 digits.
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.
Course 2
3-1 Prime Factorization
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.
Course 2
3-1 Prime Factorization
Additional Example 1A: 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.
Course 2
3-1 Prime Factorization
Additional 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.
Course 2
3-1 Insert
Lesson Title Here
Prime Factorization
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.
Course 2
3-1 Insert
Lesson Title Here
Prime Factorization
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.
Course 2
3-1 Prime Factorization
You can also use a step diagram to find the
prime factorization of a number. At each
step, divide by the smallest possible prime
number. Continue dividing until the quotient
is 1. The prime factors are the number are
the prime numbers you divided by.
Course 2
3-1 Prime Factorization
Additional Example 2A: 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.
Course 2
3-1 Prime Factorization
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.
Course 2
3-1 Insert
Lesson Title Here
Prime Factorization
Try This: Example 2A
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.
Course 2
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
3-1 Insert
Lesson Title Here
Prime Factorization
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.
Course 2
3-1 Prime Factorization
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.
Course 2
3-1 Prime
Insert Factorization
Lesson Title Here
Lesson Quiz
Use a factor tree to find the prime factorization.
33
1. 27
2. 36
2 2 · 32
3. 28
22 · 7
Use a step diagram to find the prime factorization.
4. 132
5. 52
6. 108
Course 2
22 · 3 · 11
22 · 13
22 · 33