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3-1 Prime Factorization
Preview
Warm Up
California Standards
Lesson Presentation
Holt CA Course 1
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
Holt CA Course 1
3-1 Prime Factorization
California
Standards
Preparation for
NS2.4
Determine the least common multiple and
the greatest common divisor of whole
numbers; use them to solve problems with
fractions (e.g. to find a common denominator
to add two fractions or to find the reduced
form for a fraction).
Holt CA Course 1
3-1 Prime Factorization
Vocabulary
prime number
composite number
prime factorization
Holt CA Course 1
3-1 Prime Factorization
A prime number is a whole number greater
than 1 that has exactly two positive factors, 1
and itself. Three is a prime number because its
only positive factors are 1 and 3.
A composite number is a whole number that
has more than two positive factors. Six is a
composite number because it has more than
two positive factors—1, 2, 3, and 6. The
number 1 has exactly one positive factor and
is neither prime nor composite.
Holt CA Course 1
3-1 Prime Factorization
Additional Example 1: Identifying Prime and
Composite Numbers
Tell whether each number is prime or composite.
A. 11
The positive factors
of 11 are 1 and 11.
B. 16
The positive factors of
16 are 1, 2, 4, 8, and
16.
11 is prime.
16 is composite.
Holt CA Course 1
3-1 Prime Factorization
Check It Out! Example 1
Tell whether each number is prime or composite.
A. 14
B. 7
The positive factors
of 14 are 1, 2, 7, and
14.
The positive factors
of 7 are 1 and 7.
14 is composite.
7 is prime.
Holt CA Course 1
3-1 Prime Factorization
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.
Holt CA Course 1
3-1 Prime Factorization
Writing Math
You can write prime factorization by using
exponents. The exponent tells how many times
to use the base as a factor.
Holt CA Course 1
3-1 Prime Factorization
Additional Example 2A: Using a Factor Tree to Find
Prime Factorization
Write the prime factorization of the number.
24
24
8  3
4  2 3
2  2  2  3
Write 24 as the product of
two positive factors.
Continue factoring until all
factors are prime.
The prime factorization of 24 is 2  2  2  3
or 23  3.
Holt CA Course 1
3-1 Prime Factorization
Additional Example 2B: Using a Factor Tree to Find
Prime Factorization
Write the prime factorization of the number.
150
150
30  5
10  3  5
2 5  3  5
Write 150 as the product
of two positive factors.
Continue factoring until
all factors are prime.
The prime factorization of 150 is 2  3  5  5, or
2  3  52.
Holt CA Course 1
3-1 Prime Factorization
Check It Out! Example 2A
Write the prime factorization of the number.
225
225
45  5
9  5 5
Write 225 as the product
of two positive factors.
Continue factoring until all
factors are prime.
3  3  5  5
The prime factorization of 225 is 3  3  5  5,
or 32  52.
Holt CA Course 1
3-1 Prime Factorization
Check It Out! Example 2B
Write the prime factorization of the number.
90
90
45  2
9  5  2
3 3  5  2
Write 90 as the product
of two positive factors.
Continue factoring until
all factors are prime.
The prime factorization of 90 is 3  3  5  2, or
2  32  5.
Holt CA Course 1
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.
Holt CA Course 1
3-1 Prime Factorization
Additional Example 3A: Using a Step Diagram to Find
Prime Factorization
Write the prime factorization of the number.
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.
Holt CA Course 1
3-1 Prime Factorization
Additional Example 3B: Using a Step Diagram to Find
Prime Factorization
Write the prime factorization of the number.
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.
Holt CA Course 1
3-1 Prime Factorization
Check It Out! Example 3A
Write the prime factorization of the number.
324
2 324
2 162
3 81
3 27
3 9
3 3
1
The prime factorization
or 22  34.
Holt CA Course 1
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,
3-1 Prime Factorization
Check It Out! Example 3B
Write the prime factorization of the number.
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.
Holt CA Course 1
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.
Holt CA Course 1
3-1 Prime Factorization
Lesson Quiz
Tell whether each number is prime or composite.
1. 23 prime
2. 39 composite 3. 27 composite
Write the prime factorization of each number.
4. 27 33
5. 36 22  32
6. 28 22  7
7. 132 22  3  11
8. 52 22  13
Holt CA Course 1
9. 108 22  33