Factors and Simplest Form Section 4.2 b A prime number is a natural number greater than 1 whose only factors are 1 and itself. The first few prime numbers

Factors and Simplest Factors and Simplest FormForm

Section 4.2

A A prime numberprime number is a natural number is a natural number greater than 1 whose only factors are greater than 1 whose only factors are 1 and itself. The first few prime 1 and itself. The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, . . .23, 29, . . .

A A composite numbercomposite number is a natural is a natural number greater than 1 that is not number greater than 1 that is not prime.prime.

2Martin-Gay, Prealgebra, 5ed

The natural number 1 is neither The natural number 1 is neither prime nor composite. prime nor composite.

Helpful Hint


Prime FactorizationPrime Factorization

A A prime factorizationprime factorization of a number of a number expresses the number as a expresses the number as a product product of its of its factorsfactors and the and the factors must be prime numbers.factors must be prime numbers.


Remember aRemember a factorfactor is any is any number that divides a number number that divides a number evenly (with a remainder of 0).evenly (with a remainder of 0).

Helpful Hint


Every whole number greater than 1 has Every whole number greater than 1 has exactly one prime factorization.exactly one prime factorization.

12 = 2 • 2 • 3

2 and 3 are 2 and 3 are prime factorsprime factors of 12 because of 12 because they are prime numbers and they divide they are prime numbers and they divide evenly into 12.evenly into 12.


Divisibility TestsDivisibility Tests

A whole number is divisible byA whole number is divisible by 2 if its last digit is 0, 2, 4, 6, or 8.2 if its last digit is 0, 2, 4, 6, or 8.

3 if the sum of its digits is divisible by 3.3 if the sum of its digits is divisible by 3.

196 is divisible by 2

117 is divisible by 3 since 1 + 1 + 7 = 9 is divisible by 3.


Divisibility TestsDivisibility Tests

A whole number is divisible byA whole number is divisible by 5 if the ones digit is 0 or 5.5 if the ones digit is 0 or 5.

10 if its last digit is 0.10 if its last digit is 0.

2,345 is divisible by 5.

8,470 is divisible by 10.


Equivalent FractionsEquivalent Fractions

Graph on the number line.Graph on the number line.34 3

4

Graph on the number line.Graph on the number line.68

68

00 11

1

41

8


Fractions that represent the Fractions that represent the same portion of a whole or the same portion of a whole or the same point on the number line same point on the number line are calledare called equivalent fractionsequivalent fractions..

10


Martin-Gay, Prealgebra, 5ed

6 2 3

8 2 4

68


3 2 6

4 2 8

34


Fundamental Property of Fundamental Property of FractionsFractions

If If aa, , bb, and , and cc are numbers, then are numbers, then

and also and also

a a c

b b c

a a c

b b c

as long as as long as bb and and cc are not 0. If the are not 0. If the numerator and denominator are numerator and denominator are multiplied or divided by the multiplied or divided by the samesame nonzero number, the result is an nonzero number, the result is an equivalent fractionequivalent fraction. .


Simplest FormSimplest Form

A fraction is in A fraction is in simplest formsimplest form, or , or lowest termslowest terms, , when the numerator and denominator have no when the numerator and denominator have no common factors other than 1.common factors other than 1.

14 14 7 2

21 21 7 3

Using the fundamental principle of fractions, divide the numerator and denominator by the common factor of 7.

Using the prime factorization of the numerator and denominator, divide out common factors.

14 7 2

21 7 3

2

3

7 2

7 313

Martin-Gay, Prealgebra, 5ed

Writing a Fraction in Simplest Writing a Fraction in Simplest FormForm

To write a fraction in simplest form, write the prime factorization of the 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.numerator and the denominator and then divide both by all common factors.

The process of writing a fraction in simplest form is called The process of writing a fraction in simplest form is called simplifyingsimplifying the the fraction.fraction.


When all factors of the numerator or When all factors of the numerator or denominator are divided out, don’t denominator are divided out, don’t forget that 1 still remains in that forget that 1 still remains in that numerator or denominator.numerator or denominator.

Helpful Hint

5 5 1

10 5 2 2

15 3 5 5

53 3 1


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Factors and Simplest Form Section 4.2 b A prime number is a natural number greater than 1 whose only factors are 1 and itself. The first few prime numbers