Download 2-6 Prime Factorization

2-6 Prime Factorization
Learn to find the prime factorizations of
composite numbers.
2-6 Prime Factorization
Vocabulary
prime number
composite number
prime factorization
2-6 Prime Factorization
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.
2-6 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.
2-6 Prime Factorization
Additional Example 1: Identifying Prime and
Composite Numbers
Tell whether each number is prime or composite.
A. 11
The factors of 11 are
1 and 11.
11 is prime.
B. 16
The factors of 16 are
1, 2, 4, 8, and 16.
16 is composite.
2-6 Prime Factorization
Check It Out: Example 1
Tell whether each number is prime or composite.
A. 14
B. 7
The factors of 14 are
1, 2, 7, and 14.
The factors of 7 are 1
and 7.
14 is composite.
7 is prime.
2-6 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.
2-6 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.
2-6 Prime Factorization
Additional Example 2A: Using a Factor Tree to Find
Prime Factorization
Write the prime factorization of each number.
24
24
Write 24 as the product of
two factors.
8 · 3
4 · 2 · 3
Continue factoring until all
factors are prime.
2 · 2 · 2 · 3
The prime factorization of 24 is 2 · 2 · 2 · 3
or 23 · 3.
2-6 Prime Factorization
Additional Example 2B: Using a Factor Tree to Find
Prime Factorization
Write the prime factorization of each number.
150
150
30 · 5
10 · 3 · 5
Write 150 as the product
of two factors.
Continue factoring until
all factors are prime.
2 · 5 · 3 · 5
The prime factorization of 150 is 2 · 3 · 5 · 5, or
2 · 3 · 52.
2-6 Prime Factorization
Check It Out: Example 2A
Write the prime factorization of each number.
36
36
Write 36 as the product of
two factors.
18 · 2
9 · 2 · 2
Continue factoring until all
factors are prime.
3 · 3 · 2 · 2
The prime factorization of 36 is 2 · 2 · 3 · 3 or
22 · 32.
2-6 Prime Factorization
Check It Out: Example 2B
Write the prime factorization of the number.
90
90
45 · 2
9 · 5 · 2
Write 90 as the product
of two factors.
Continue factoring until
all factors are prime.
3 · 3 · 5 · 2
The prime factorization of 90 is 3 · 3 · 5 · 2, or
2 · 32 · 5.
2-6 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.
2-6 Prime Factorization
Additional Example 3A: Using a Step Diagram to Find
Prime Factorization
Write the prime factorization of each 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.
2-6 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.
2-6 Prime Factorization
Check It Out: Example 3A
Write the prime factorization of each number.
324
2 324
2 162
3 81
3 27
3 9
3 3
1
The prime factorization
22 · 34.
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
2-6 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.
2-6 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.