Transcript Number Theory: Prime and Composite Numbers

CHAPTER 5
Number Theory and the
Real Number System
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5.1
Number Theory: Prime & Composite
Numbers
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Objectives
1. Determine divisibility.
2. Write the prime factorization of a composite
number.
3. Find the greatest common divisor of two
numbers.
4. Solve problems using the greatest common
divisor.
5. Find the least common multiple of two numbers.
6. Solve problems using the least common
multiple.
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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,...
• Natural numbers that are multiplied together are
called the factors of the resulting product.
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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 number 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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Example 2: 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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Example 2 (continued)
Thus, the prime factorization of 700 is
700 = 7  2  2  5  5
= 7  22  52
2
2
 2 5 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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Greatest Common Divisor
1.
2.
3.
•
To find the greatest common divisor of two or more
numbers,
Write the prime factorization of each number.
Select each prime factor with the smallest exponent
that is common to each of the prime factorizations.
Form the product of the numbers from step 2. The
greatest common divisor is the product of these
factors.
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.
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Example 3: 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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Example 3: (continued)
216 = 23  33
234 = 2  32  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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Example 3: (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  32 = 2  9 =
18. Thus, the greatest common factor for 216 and 234
is 18.
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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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Example 5: 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  32
300 = 22  3  52
Step 2. Select every prime factor that occurs, raised to
the greatest power to which it occurs, in these
factorizations. 144 = 24  32
300 = 22  3  52
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Example 5: continued
Step 3. Form the product of the numbers from step 2.
The least common multiple is the product of these
factors.
LCM = 24  32  52 = 16  9  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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Example 6: Solving a Problem Using the Least
Common Multiple
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:
Movie 1 4:00 + 80 | 5:20 + 80 | 6:40 + 80 | 8:00
Movie 2 4:00 + 120 | 6:00 + 120 | 8:00
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Example 6: Solving a Problem Using the Least
Common Multiple
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 again.
What is the time duration (minutes) into which 80 and
120 will divide evenly?
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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Example 6: (continued)
Begin with the prime factorization of 80 and 120:
80 = 24  5
120 = 23  3  5
Now select each prime factor, with the greatest
exponent from each factorization.
LCM = 24  3  5 = 16  3  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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