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Math 64
4.2 "Factors and Simplest Form"
Bibiana Lopez
Riverside City College
June 2010
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Objectives:
* Write a number as a product of prime numbers.
* Write a fraction in simplest form.
* Determine whether two fractions are equivalent.
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Writing a Number as a Product of Prime Numbers
A factor is any number that divides a number evenly (with a
remainder of 0). For example,
12 = 2 6 or 12 = 3 4 or 12 = 2 2 3 .
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Writing a Number as a Product of Prime Numbers
A factor is any number that divides a number evenly (with a
remainder of 0). For example,
12 = 2 6 or 12 = 3 4 or 12 = 2 2 3 .
A prime number is a natural number greater than 1 whose only
factors are 1 and itself. For example,
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, . . . .
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Writing a Number as a Product of Prime Numbers
A factor is any number that divides a number evenly (with a
remainder of 0). For example,
12 = 2 6 or 12 = 3 4 or 12 = 2 2 3 .
A prime number is a natural number greater than 1 whose only
factors are 1 and itself. For example,
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, . . . .
A composite number is a natural number greater than 1 that is
not prime. For example,
4, 6, 8, 9, 10, 12, 14, 15, 16, 18, . . .
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Writing a Number as a Product of Prime Numbers
The number 1 is neither prime nor composite. When a composite
number is written as a product of prime numbers, this product is called
the prime factorization of the number. For example, 12 = 2 2 3 .
One method for …nding the prime factorization of a number is by using a
factor tree.
Example 1: (Writing a number as a product of prime numbers)
Use a factor tree to …nd the prime factorization of each number.
a) 30
b) 56
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Writing a Number as a Product of Prime Numbers
c) 72
d) 75
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Writing a Number as a Product of Prime Numbers
Divisibility Tests:
A whole number is divisible by:
2 if the last digit is 0, 2, 4, 6, or 8
.
3
if the sum of the digits is divisible by 3
5
if the last digit is 0 or 5
4
if its last two digits are divisible by 4
6
if it is divisible by 2 and 3
9
if the sum of its digits is divisible by 9
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Writing a Number as a Product of Prime Numbers
Example 2: (Writing a number as a product of prime numbers)
Write the prime factorization of each number.
a) 297
b) 836
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Writing Fractions in Simplest Form
Fractions that represent the same portion of a whole or the same
point on a number line are called equivalent fractions. For example,
Simplest Form of a Fraction
A fraction is written in simplest form or lowest terms when the numerator and the denominator have no common factors other than 1.
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Writing Fractions in 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.
Note:
hn
i
Any nonzero number n divided by itself is 1
=1 .
n
5
17
24
For example,
= 1,
= 1, or
=1
5
17
24
Property of Fractions
c
a
If and are fractions (with b and d not 0), the following is true.
b
d
a c
a c
=
b d
b d
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Writing Fractions in Simplest Form
Example 3: (Writing fractions in simplest form)
Write each fraction in simplest form.
30
49
a)
b)
45
112
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Writing Fractions in Simplest Form
c)
7a2
56a3
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d)
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40xy
64xyz
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Determining Whether Two Fractions are Equivalent
Recall that two fractions are equivalent if they represent the same
part of a whole. One way to determine whether two fractions are
equivalent is to see whether they simplify to the same fraction.
Example 4: (Determining whether two fractions are equivalent)
Determine whether each pair of fractions is equivalent.
7
6
14
2
and
b)
and
a)
8
28
21
35
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Determining Whether Two Fractions are Equivalent
There is a shortcut method we may use to check whether two
fractions are equivalent called cross products.
Example 5: (Determining whether two fractions are equivalent)
Determine whether each pair of fractions is equivalent.
6
2
4
10
and
b)
and
a)
15
9
5
11
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