Chapter 3 Fractions

Factors and Simplest Form 3.2 Factors and Simplest Form

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

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

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

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

Examples Find the prime factorization of 30. Write 30 as the product of two numbers. Continue until all factors are prime. 30 6 • 5 3 • 2 • 5 The prime factorization of 30 is 2 · 3 · 5. Objective A 7

Examples Find the prime factorization of 36. Write 36 as the product of two numbers. Continue until all factors are prime. 36 9 • 4 3 • 3 2 • 2 The prime factorization of 36 is 3 · 3 · 2 · 2 or 32 · 22. Objective A 8

Divisibility Tests Objective A 9

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

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

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

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

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

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

Equality of Fractions Objective A 16

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

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