1. 3-1 Prime Factorization Warm Up 1. 6 2. 16 3. 17 4. 36 1 · 6, 2 · 3
Course 2
3-1
Prime Factorization
Warm Up
Write each number as a product of two whole numbers in as many ways as possible.
1. 6
2. 16
3. 17
4. 36
5. 23
1 · 6, 2 · 3
1 · 16, 2 · 8, 4 · 4
1 · 17
1 · 36, 2 · 18, 3 · 12, 4 · 9, 6 · 6
1 · 23

2. Course 2
3-1
Prime Factorization
Learn to find the prime factorizations of composite numbers.

3. Insert Lesson Title Here
Course 2
3-1
Prime Factorization
Insert Lesson Title Here
Vocabulary
prime number
composite number
prime factorization

4. 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.

5. 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.

6. Additional Example 1A: Using a Factor Tree to Find Prime Factorization
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
Write 24 as the product of
two factors. 24
8 · 3
Continue factoring until all
factors are prime.
4 · 2 · 3
2 · 2 · 2 · 3
The prime factorization of 24 is 2 · 2 · 2 · 3. Using
exponents, you can write this as 23 · 3.

7. Additional Example B: Using a Factor Tree to Find Prime Factorization
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
Write 150 as the product
of two factors.
30 · 5
Continue factoring until
all factors are prime.
10 · 3 · 5
2 · 5 · 3 · 5
The prime factorization of 150 is 2 · 3 · 5 · 5, or
2 · 3 · 52.

8. Insert Lesson Title Here
Course 2
3-1
Prime Factorization
Insert Lesson Title Here
Try This: Example 1A
Write the prime factorization of the number.
A. 36
Write 36 as the product of
two factors. 36
18 · 2
Continue factoring until all
factors are prime.
9 · 2 · 2
3 · 3 · 2 · 2
The prime factorization of 36 is 2 · 2 · 3 · 3. Using
exponents, you can write this as 22 · 32.

9. Insert Lesson Title Here
Course 2
3-1
Prime Factorization
Insert Lesson Title Here
Try This: Example 1B
Write the prime factorization of the number.
B. 90 90
Write 90 as the product
of two factors.
45 · 2
Continue factoring until
all factors are prime.
9 · 5 · 2
3 · 3 · 5 · 2
The prime factorization of 90 is 3 · 3 · 5 · 2, or
2 · 32 · 5.

10. 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.

11. 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
Divide 476 by 2. Write the
quotient below 476. 2
476 2
238
Keep dividing by a prime
number. 7
119 17 17 1
Stop when the quotient is 1.
The prime factorization of 476 is 2 · 2 · 7 · 17, or
22 · 7 · 17.

12. 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
Divide 275 by 5. Write the quotient
below 275. 5
275 5 55 11 11
Stop when the quotient is 1. 1
The prime factorization of 275 is 5 · 5 · 11, or
52 · 11.

13. Insert Lesson Title Here
Course 2
3-1
Prime Factorization
Insert Lesson Title Here
Try This: Example 2A
Write the prime factorization of each number.
A. 324
Divide 324 by 2. Write the
quotient below 324. 2
324 2
162
Keep dividing by a prime
number. 3 81 3 27 3 9
Stop when the quotient is 1. 3 3 1
The prime factorization of 324 is 2 · 2 · 3 · 3 · 3 · 3, or
22 · 34.

14. Insert Lesson Title Here
Course 2
3-1
Prime Factorization
Insert Lesson Title Here
Try This: Example 2B
Write the prime factorization of the number.
B. 325
Divide 325 by 5. Write the quotient
below 325. 5
325 5 65 13 13
Stop when the quotient is 1. 1
The prime factorization of 325 is 5 · 5 · 13, or
52 · 13.

15. 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 7
476 2
238 2 68 7
119 2 34 17 17 17 17 1 1
The prime factorizations are 2 · 2 · 7 · 17 and
7 · 2 · 2 · 17, which are the same as 17 · 2 · 2 · 7.

16. Insert Lesson Title Here
Course 2
3-1
Prime Factorization
Insert Lesson Title Here
Lesson Quiz
Use a factor tree to find the prime factorization.
1. 27
2. 36
3. 28
Use a step diagram to find the prime factorization.
4. 132
5. 52
6. 108 33
22 · 32
22 · 7
22 · 3 · 11
22 · 13
22 · 33