1. Chapter 3
Fractions

2. Factors and Simplest Form
3.2
Factors and Simplest Form

3. 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 3

4. Prime and Composite Numbers
Prime Numbers
A prime number is a natural number that has exactly two different factors 1 and itself.
Composite Numbers
A composite number is any natural number, other than 1, that is not prime.
Objective A 4

5. Examples
Determine whether each number is prime or composite. Explain your answers.
a. 16
b. 31
c. 49
Composite, it has more than two factors: 1, 2, 4, 8, 16.
Prime, its only factors are 1 and 31.
Objective A
Composite, it has more than two factors: 1, 7, 49. 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 Find 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. 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 11

12. Examples
Write in simplest form. a. b. c.
Objective A 12

13. 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 13

14. Examples
Write in simplest form. a. b. c.
Objective A 14

15. Example Determine whether are equivalent. Simplify each fraction.
Since both of the simplified fractions are the same, they are equivalent.
Objective A 15

16. Equality of Fractions
Objective A 16

17. Example Determine whether are equivalent by cross multiplying.
Objective A
Since 22 ≠20, then 17

18. Solving Problems by Writing Fractions in Simplest Form
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