Holt CA Course 1

3-1 Prime Factorization

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).

California Standards

Holt CA Course 1

3-1 Prime FactorizationVocabulary

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.

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

Tell whether each number is prime or composite.

Example 1: Identifying Prime and Composite Numbers

A. 11

So 11 is prime.

The positive factors of 11 are 1 and 11.

B. 16

So 16 is composite.

The positive factors of 16 are 1, 2, 4, 8, and 16.

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

You can write prime factorization by using exponents. The exponent tells how many times to use the base as a factor.

Writing Math

Holt CA Course 1

3-1 Prime Factorization

Write the prime factorization of the number.

Example 2A: Using a Factor Tree to Find Prime Factorization

2424

8 3

4 2 3

2 2 2 3

Write 24 as the product oftwo positive factors.

Continue factoring until allfactors are prime.

The prime factorization of 24 is 2 2 2 3 . Using exponents, you can write this as 23 3.

Holt CA Course 1

3-1 Prime Factorization

Write the prime factorization of the number.

Example 2B: Using a Factor Tree to Find Prime Factorization

150150

30 5

10 3 5

2 5 3 5

Write 150 as the productof two positive factors.

Continue factoring until all factors are prime.

The prime factorization of 150 is 2 3 5 5, or2 3 52.

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

Write the prime factorization of the number.

Example 3A: Using a Step Diagram to Find Prime Factorization

476

476238

119171

22

717

Divide 476 by 2. Write the quotient below 476.

Keep dividing by a prime factor.

Stop when the quotient is 1.

The prime factorization of 476 is 2 2 7 17, or22 7 17.

Holt CA Course 1

3-1 Prime Factorization

Write the prime factorization of the number.

Example 3B: Using a Step Diagram to Find Prime Factorization

275

27555111

5511

Divide 275 by 5. Write the quotientbelow 275.

Stop when the quotient is 1.

The prime factorization of 275 is 5 5 11, or52 11.

Keep dividing by a prime factor.

Holt CA Course 1

3-1 Prime Factorization

Tell whether the number is prime or composite.

Check It Out! Example 1

14

14 is composite.

The positive factors of 14 are 1, 2, 7, and 14.

Holt CA Course 1

3-1 Prime FactorizationCheck It Out! Example 2

Write the prime factorization of the number.

9090

45 2

9 5 2

3 3 5 2

Write 90 as the productof two positive factors.

Continue factoring until all factors are prime.

The prime factorization of 90 is 3 3 5 2, or2 32 5.

Holt CA Course 1

3-1 Prime Factorization

Check It Out! Example 3

Write the prime factorization of the number.

325

32565131

5513

Divide 325 by 5. Write the quotientbelow 325.

Stop when the quotient is 1.

The prime factorization of 325 is 5 5 13, or52 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.

476238119

171

22

717

4766834

171

72

217

The prime factorizations are 2 2 7 17 and7 2 2 17, which are the same as 17 2 2 7.