Download Study Guide 5-1 thru 5-4 Prime Factorization (5-1) - shs

Study Guide
5-1 thru 5-4
Prime Factorization (5-1)
A prime number is a whole number greater than 1 that has exactly two factors, 1 and itself. A
whole number greater than 1 that has more than two factors is a composite number. Every
composite number can be written as the product of prime numbers in exactly one way. This is
called the prime factorization of a number.
Example 1 – Determine whether 21 is prime or composite.
The number 21 has four factors: 1, 3, 7, and 21. So it is composite.
Example 2 – Find the prime factorization of 420
Using the factor tree
Using the “L” method
2 420
2 210
3 105
5 35
7
The prime factorization of
420 is 2 x 2 x 5 x 3 x 7
Divide 420 by the lowest prime number.
Keep dividing until you have reached a prime
number.
Keep factoring until all of the factors are prime
List the vertical and horizontal numbers to get
numbers.
your prime factorization.
The prime factorization of 420 is 2 x 2 x 5 x 3 x 7, or 22 x 3 x 5 x 7.
Write 420 as the product of the two factors.
Greatest Common Factor (5-2)
The greatest common factor (GCF) of two or more numbers is the largest number that is a
factor of each number. The GCF of prime numbers is 1.
Example 1 – Find the GCF of 72 and 108 by listing factors.
Factors of 72:
1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
Factors of 108:
1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108
Common Factors:
1, 2, 3, 4, 6, 9, 12, 18, 36
The GCF of 72 and 108 is 36
Example 2 – Find the GCF of 42 and 60 using prime factors.
Method 1 – Write the prime factorization
Method 2 – Divide by prime numbers.
Divide 42 and 60 by 2
Then divide the quotients by 3
2 42 60
60 =
42 =
2
×
2
×
3
×
5
3 21 30
2
×
3
×
7
7
15
GCF = 2 × 3 or 6
The common prime factors are 2 and 3. The GCF of 42 and 60 is 2 × 3, or 6.
Simplifying Fractions (5-3)
Fractions that have the same value are called equivalent fractions. A fraction is in simplest form
when the GCF of the numerator and the denominator is 1.
36
in simplest form
54
First, find the GCF of the numerator and denominator.
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
Factors of 54: 1, 2, 3, 6, 9, 18, 27, 54
The GCF of 36 and 54 is 18.
Example 1 - Write
Then, divide the numerator and the denominator by the GCF.
36
36 ÷ 18
2
36
=
=
So,
written in simplest form is
54
54 ÷ 18
3
54
Example 2 - Write
8
12
2
.
3
in simplest form
8 = 2 × 2 × 2
12 = 2 × 2 × 3
=
12
2 4
6
2
GCF: 2 × 2 = 4
8
8÷4
2
=
=
12
12 ÷ 4
3
2 8
3
The 2 becomes the numerator
and the 3 becomes the
denominator of the fraction
8
2
=
12
3
Fractions and Decimals (5-4)
To write a decimal as a fraction, divide the numerator of the fraction by the denominator. Use a
power of ten to change a decimal to a fraction.
5
as a decimal.
9
Method 1 – Use pencil and paper.
0.555…
9 5.000
45
The remainder after
50
each step is 5.
45
50
45
5
Example 1 - Write
Method 2 – Use a calculator
5
÷
9
=
0.55555556
You can use bar notation 0.5 to indicate the 5
repeats forever.
5
So,
= 0. 5
9
Example 2 – Write 0.32 as a fraction in simplest form.
32
0.32 =
The 2 is in the hundredths place.
100
=
8
25
Simplify.