Warm Up

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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
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Learn to find the prime factorizations of
composite numbers.
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Insert Lesson Title Here
Vocabulary
prime number composite number prime factorization
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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.
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A composite number is a whole number that has
more than two factors. Six is a composite number
because it has more than two factors1, 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.
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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.
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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.
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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.
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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.
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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.
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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.
476
2
238
2
Keep dividing by a prime number.
119
7
17
17
1
Stop when the quotient is 1.
The prime factorization of 476 is 2 2 7 17,
or 22 7 17.
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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.
275
5
55
5
11
11
Stop when the quotient is 1.
1
The prime factorization of 275 is 5 5 11,
or 52 11.
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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.
324
2
162
2
Keep dividing by a prime number.
81
3
27
3
9
3
Stop when the quotient is 1.
3
3
1
The prime factorization of 324 is 2 2 3 3
3 3, or 22 34.
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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.
325
5
65
5
13
13
Stop when the quotient is 1.
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The prime factorization of 325 is 5 5 13,
or 52 13.
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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.
476
476
2
7
238
68
2
2
119
34
7
2
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.
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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
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22 32
22 7
22 3 11
22 13
22 33