Section 5.1
Number Theory: Prime & Composite Numbers
1.
2.
3.
4.
5.
6.
Objectives
Determine divisibility.
Write the prime factorization of a composite number.
Find the greatest common divisor of two numbers.
Solve problems using the greatest common divisor.
Find the least common multiple of two numbers.
Solve problems using the least common multiple.
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Section 5.1
1
Number Theory and Divisibility
• Number theory is primarily concerned with the
properties of numbers used for counting, namely 1, 2, 3,
4, 5, and so on.
• The set of natural numbers is given by
N = {1,2,3,4,5,6,7,8,9,10,11,...}
• The natural numbers that are multiplied are called the
factors of the product.
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2
Number Theory and Divisibility
• If a and b are natural numbers, a is divisible by b if the
operation of dividing a by b leaves a remainder of 0.
• This is the same as saying that b is a divisor of a, or b
divides a.
• This is symbolized by writing b|a.
Example: We write 12|24 because 12 divides 24 or 24
divided by 12 leaves a remainder of 0. Thus, 24 is
divisible by 12.
Example: If we write 13|24, this means 13 divides 24 or 24
divided by 13 leaves a remainder of 0. But this is not
true, thus, 13|24.
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Prime Factorization
• A prime number is a natural number greater than 1 that
has only itself and 1 as factors.
• A composite number is a natural greater than 1 that is
divisible by a number other than itself and 1.
• The Fundamental Theorem of Arithmetic
Every composite number can be expressed as a
product of prime numbers in one and only one
way.
• One method used to find the prime factorization of a
composite number is called a factor tree.
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Section 5.1
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Prime Factorization
Prime Factorization using a Factor Tree
Example: Find the prime factorization of 700.
Solution: Start with any two numbers whose product is 700,
such as 7 and 100.
Continue factoring the
composite number, branching
until the end of each branch
contains a prime number.
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Section 5.1
5
Prime Factorization
Prime Factorization using a Factor Tree
Example Continued
Thus, the prime factorization of 700 is
700 = 7 x 2 x 2 x 5 x 5
= 7 x 22 x 52
= 2 2 ⋅ 52 ⋅ 7
Notice, we rewrite the prime factorization using a dot to
indicate multiplication, and arranging the factors from
least to greatest.
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Section 5.1
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Greatest Common Divisor
•
Pairs of numbers that have 1 as their greatest common
divisor are called relatively prime.
For example, the greatest common divisor of 5 and 26 is 1.
Thus, 5 and 26 are relatively prime.
To find the greatest common divisor of two or more
numbers,
1. Write the prime factorization of each number.
2. Select each prime factor with the smallest exponent that
is common to each of the prime factorizations.
3. Form the product of the numbers from step 2. The
greatest common divisor is the product of these factors.
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Greatest Common Divisor
Finding the Greatest Common Divisor
Example: Find the greatest common divisor of 216 and
234.
Solution: Step 1. Write the prime factorization of each
number.
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8
Greatest Common Divisor
Example Continued
The factor tree at the left indicates that 216 = 23 x 33.
The factor tree at the right indicates that 234 = 2 x 32 x 13.
Step 2. Select each prime factor with the smallest exponent
that is common to each of the prime factorizations.
Which exponent is appropriate for 2 and 3? We choose the
smallest exponent; for 2 we take 21, for 3 we take 32.
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Greatest Common Divisor
Example Continued
Step 3. Form the product of the numbers from step 2. The
greatest common divisor is the product of these factors.
Greatest common divisor = 2 x 32 = 2 x 9 = 18. Thus, the
greatest common factor for 216 and 234 is 18.
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Section 5.1
10
Least Common Multiple
•
The least common multiple of two or more natural
numbers is the smallest natural number that is divisible
by all of the numbers.
To find the least common multiple using prime factorization
of two or more numbers:
1. Write the prime factorization of each number.
2. Select every prime factor that occurs, raised to the
greatest power to which it occurs, in these
factorizations.
3. Form the product of the numbers from step 2. The least
common multiple is the product of these factors.
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Least Common Multiple
Finding the Least Common Multiple
Example: Find the least common multiple of 144 and 300.
Solution: Step 1. Write the prime factorization of each
number.
144 = 24 x 32
300 = 22 x 3 x 52
Step 2. Select every prime factor that occurs, raised to the
greatest power to which it occurs, in these factorizations.
144 = 24 x 32
300 = 22 x 3 x 52
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Least Common Multiple
Example Continued
Step 3. Form the product of the numbers from step 2. The
least common multiple is the product of these factors.
LCM = 24 x 32 x 52 = 16 x 9 x 25 = 3600
Hence, the LCM of 144 and 300 is 3600. Thus, the
smallest natural number divisible by 144 and 300 is
3600.
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Least Common Multiple
Solving a Problem using the LCM
Example: A movie theater runs its films continuously. One
movie runs for 80 minutes and a second runs for 120
minutes. Both movies begin at 4:00 p.m. When will the
movies begin again at the same time?
Solution: We are asked to find when the movies will begin
again at the same time. Therefore, we are looking for the
LCM of 80 and 120. Find the LCM and then add this
number of minutes to 4:00 p.m.
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Least Common Multiple
Solving a Problem using the LCM
Example Continued
Begin with the prime factorization of 80 and 120:
80 = 24 x 5
120 = 23 x 3 x 5
Now select each prime factor, with the greatest exponent
from each factorization.
LCM = 24 x 3 x 5 = 16 x 3 x 5 = 240
Therefore, it will take 240 minutes, or 4 hours, for the
movies to begin again at the same time. By adding 4
hours to 4:00 p.m., they will start together again at 8:00
p.m.
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Section 5.7
Arithmetic and Geometric Sequences
1.
2.
3.
4.
Objectives
Write terms of an arithmetic sequence.
Use the formula for the general term of an arithmetic
sequence.
Write terms of a geometric sequence.
Use the formula for the general term of a geometric
sequence.
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Section 5.7
1
Sequences
• A sequence is a list of numbers that are related to each
other by a rule.
• The numbers in the sequence are called its terms.
For example, a Fibonacci sequence term takes the sum of
the two previous successive terms, i.e.,
1+1=2
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1+2=3
Section 5.7
3+2=5
5+3=8
2
Arithmetic Sequences
• An arithmetic sequence is a sequence in which each
term after the first differs from the preceding term by a
constant amount.
• The difference between consecutive terms is called the
common difference of the sequence.
Arithmetic Sequence
Common Difference
142, 146, 150, 154, 158, …
d = 146 – 142 = 4
-5, -2, 1, 4, 7, …
d = -2 – (-5) = -2 + 5 = 3
8, 3, -2, -7, -12, …
d = 3 – 8 = -5
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Arithmetic Sequences
Write the Terms of an Arithmetic Sequence
Example: Write the first six terms of the arithmetic
sequence with first term 6 and common difference 4.
Solution: The first term is 6. The second term is 6 + 4 = 10.
The third term is 10 + 4 = 14, and so on. The first six
terms are
6, 10, 14, 18, 22, and 26
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The General Term of an Arithmetic Sequence
• Consider an arithmetic sequence with first term a1. Then
the first six terms are
• Using the pattern of the terms results in the following
formula for the general term, or the nth term, of an
arithmetic sequence:
The nth term (general term) of an arithmetic
sequence with first term a1 and common difference
d is
an = a1 + (n – 1)d
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The General Term of an Arithmetic Sequence
Example: Find the eighth term of the arithmetic sequence
whose first term is 4 and whose common difference is -7.
Solution: To find the eighth term, a8, we replace n in the
formula with 8, a1 with 4, and d with -7.
an = a1 + (n – 1)d
a8 = 4 + (8 – 1)(-7)
= 4 + 7(-7)
= 4 + (-49)
= -45
The eighth term is -45.
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Section 5.7
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Geometric Sequences
• A geometric sequence is a sequence in which each
term after the first is obtained by multiplying the
preceding term by a fixed nonzero constant.
• The amount by which we multiply each time is called the
common ratio of the sequence.
Geometric Sequence
Common Ratio
1, 5, 25, 125, 625, …
r
=
4, 8, 16, 32, 64, …
r
=
6, -12, 24, -48, 96, …
r =
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Section 5.7
5
1
8
4
− 12
6
=
5
=
2
= − 2
7
Writing the Terms of a Geometric Sequences
Example: Write the first six terms of the geometric
sequence with first term 6 and common ratio ⅓.
Solution: The first term is 6. The second term is 6 · ⅓ = 2.
The third term is 2 · ⅓ = ⅔, and so on. The first six terms
are
2
2 2 2
6 ,2 , , ,
, and
.
3 9 27
81
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Section 5.7
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The General Term of a Sequence
• Consider a geometric sequence with first term a1 and
common ration r. Then the first six terms are
• Using the pattern of the terms results in the following
formula for the general term, or the nth term, of a
geometric sequence:
The nth term (general term) of a geometric sequence
with first term a1 and common ratio r is
an = a1rn-1
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The General Term of a Sequence
Example: Find the eighth term in the geometric sequence
whose first term is -4 and whose common ratio is -2.
Solution: To find the eighth term, a8, we replace n in the
formula with 8, a1 with -4, and r with -2.
an = a1rn-1
a8 = -4(-2)8-1
= -4(-2)7
= -4(-128)
= 512
The eighth term is 512.
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Section 5.7
10