1. 3
Chapter
Chapter 2
Fractions and Mixed Numbers

2. Factors and Simplest Form
Section
3.2
Factors and Simplest Form

3. Write a Number as a Product of Prime Numbers.
Objective A
Write a Number as a Product of Prime Numbers.

4. Finding the Factors of Numbers
To perform many operations, it is necessary to be able to factor a number.
Since 7 · 9 = 63, both 7 and 9 are factors of 63, and 7 · 9 is called a factorization of 63.
Objective A 4

5. Prime and Composite Numbers
Prime Numbers
A prime number is a natural number greater than 1 whose only factors are 1 and itself. The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, … .
Composite Numbers
A composite number is a natural number, greater than 1, that is not prime.
Objective A 5

6. Prime Factorization Prime Factorization
The prime factorization of a number is the factorization in which all the factors are prime numbers.
Every whole number greater than 1 has exactly one prime factorization.
Objective A 6

7. Examples Find the prime factorization of 30.
Write 30 as the product of two numbers. Continue until all factors are prime. 30 •
3 • • 5
The prime factorization of 30 is 2 · 3 · 5.
Objective A 7

8. Examples Find the prime factorization of 36.
Write 36 as the product of two numbers. Continue until all factors are prime. 36 •
3 • • 2
The prime factorization of 36 is 3 · 3 · 2 · 2 or 32 · 22.
Objective A 8

9. Divisibility Tests
Objective A 9

10. Examples Write the prime factorization of 63.
The first prime number 2 does not divide evenly, but 3 does.
Because 21 is not prime, we divide again.
The quotient 7 is prime, so we are finished. The prime factorization of 63 is 3 · 3 · 7.
Objective A 10

11. Write a Fraction in Simplest Form.
Objective B
Write a Fraction in Simplest Form.

12. Writing Fractions in Simplest Form
Fractions that represent the same portion of a whole are called equivalent fractions.
There are many equivalent forms of a fraction. A special form of a fraction is called simplest form.
Simplest Form of a Fraction
A fraction is written in simplest form or lowest terms when the numerator and denominator have no common factors other than 1.
Objective A 12

13. Examples
Write in simplest form. a. b. c.
Objective A 13

14. Simplest Form Writing a Fraction in Simplest Form
To write a fraction in simplest form, write the prime factorization of the numerator and the denominator and then divide both by all common factors.
Objective A 14

15. Examples
Write in simplest form. a. b. c.
Objective A 15

16. Determine Whether Two Fractions Are Equivalent.
Objective C
Determine Whether Two Fractions Are Equivalent.

17. Example Determine whether are equivalent. Simplify each fraction.
Since both of the simplified fractions are the same, they are equivalent.
Objective A 17

18. Equality of Fractions
Objective A 18

19. Example Determine whether are equivalent by cross multiplying.
Objective A
Since 22 ≠20, then 19

20. Solve Problems by Writing Fractions in Simplest Form.
Objective D
Solve Problems by Writing Fractions in Simplest Form.

21. Example
There are 5280 feet in a mile. What fraction of a mile is represented by 2640.
Both 2640 and 5280 have a common factor of 2640.
Objective C
The fraction of a mile represented by 2640 is 1/2.