1. Preparation for NS 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).
California
Standards

2. Vocabulary prime number composite number prime factorization
A whole number greater than 1 that has exactly two positive factors, itself and 1.
A whole number that has more than two positive factors.
A number written as the product of its prime factors.

3. 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.

4. Example 1: Identifying Prime and Composite Numbers
Tell whether each number is prime or composite.
A. 11
B. 16
The positive factors of 11 are 1 and 11.
The positive factors of 16 are 1, 2, 4, 8, and 16.
So 11 is prime.
So 16 is composite.

5. A composite number can be written as the product of its prime factors
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. You can write prime factorization by using exponents
You can write prime factorization by using exponents. The exponent tells how many times to use the base as a factor.
Writing Math

7. Example 2A: Using a Factor Tree to Find Prime Factorization
Write the prime factorization of the number. 24 24
Write 24 as the product of
two positive 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 . Using exponents, you can write this as 23  3.

8. Example 2B: Using a Factor Tree to Find Prime Factorization
Write the prime factorization of the number.
150
150
Write 150 as the product
of two positive factors.
30  5
10  3  5
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.

9. Check It Out! Example 2A
Write the prime factorization of the number.
225
Write 225 as the product of two positive factors.
225
45  5
Continue factoring until all
factors are prime.
9  5  5
3  3  5  5
The prime factorization of 225 is 3  3  5  5, or 32  52.

10. 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.

11. Example 3A: Using a Step Diagram to Find Prime Factorization
Write the prime factorization of the number.
476
Divide 476 by 2. Write the
quotient below 476. 2
476 2
238
Keep dividing by a prime
factor. 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. Example 3B: Using a Step Diagram to Find Prime Factorization
Write the prime factorization of the number.
275
Divide 275 by 5. Write the quotient
below 275. 5
275 5 55 11 11
Keep dividing by a prime factor. 1
Stop when the quotient is 1.
The prime factorization of 275 is 5  5  11, or
52  11.

13. Check It Out! Example 3A
Write the prime factorization of the number.
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. Check It Out! Example 1
Tell whether the number is prime or composite. 14
The positive factors of 14 are 1, 2, 7, and 14.
14 is composite.

15. Check It Out! Example 2
Write the prime factorization of the number. 90 90
Write 90 as the product
of two positive factors.
45  2
9  5  2
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.

16. Check It Out! Example 3
Write the prime factorization of the number.
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.

17. There is only one prime factorization for any given composite number
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.