5.1 number theory. the study of numbers and their properties. the numbers we use to count are called...
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5.1
Number Theory
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Number Theory
The study of numbers and their properties. The numbers we use to count are called
the Natural Numbers or Counting Numbers.
{1,2,3,4,5,...}
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Factors
The natural numbers that are multiplied together to equal another natural number are called factors of the product.
Example: The factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24.
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Divisors
If a and b are natural numbers and the quotient of b divided by a has a remainder of 0, then we say that a is a divisor of b or a divides b.
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Prime and Composite Numbers
A prime number is a natural number greater than 1 that has exactly two factors (or divisors), itself and 1.
A composite number is a natural number that is divisible by a number other than itself and 1.
The number 1 is neither prime nor composite, it is called a unit.
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Rules of Divisibility
OMIT THIS PART
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The Fundamental Theorem of Arithmetic Every composite number can be written as
a unique product of prime numbers.
This unique product is referred to as the prime factorization of the number.
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Finding Prime Factorizations
Branching Method: Select any two numbers whose product is
the number to be factored. If the factors are not prime numbers, then
continue factoring each number until all numbers are prime.
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Example of branching method
Therefore, the prime factorization of
3190 = 2 • 5 • 11 • 29
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1. Divide the given number by the smallest prime number by which it is divisible.
2. Place the quotient under the given number.
3. Divide the quotient by the smallest prime number by which it is divisible and again record the quotient.
4. Repeat this process until the quotient is a prime number.
Division Method
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Write the prime factorization of 663.
The final quotient 17, is a prime number, so we stop. The prime factorization of 663 is 3 •13 •17
Example of division method
13
3
17
221
663
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Greatest Common Divisor
The greatest common divisor (GCD) of a set of natural numbers is the largest natural number that divides (without remainder) every number in that set.
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Finding the GCD
Determine the prime factorization of each number.
Find each prime factor with smallest exponent that appears in each of the prime factorizations.
Determine the product of the factors found in step 2.
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Example (GCD)
Find the GCD of 63 and 105.
63 = 32 • 7
105 = 3 • 5 • 7 Smallest exponent of each factor:
3 and 7 So, the GCD is 3 • 7 = 21
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Least Common Multiple
The least common multiple (LCM) of a set of natural numbers is the smallest natural number that is divisible (without remainder) by each element of the set.
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Finding the LCM
Determine the prime factorization of each number.
List each prime factor with the greatest exponent that appears in any of the prime factorizations.
Determine the product of the factors found in step 2.
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Example (LCM)
Find the LCM of 63 and 105.
63 = 32 • 7
105 = 3 • 5 • 7 Greatest exponent of each factor:
32, 5 and 7 So, the GCD is 32 • 5 • 7 = 315
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Example of GCD and LCM Find the GCD and LCM of 48 and 54. Prime factorizations of each:
48 = 2 • 2 • 2 • 2 • 3 = 24 • 3 54 = 2 • 3 • 3 • 3 = 2 • 33
GCD = 2 • 3 = 6
LCM = 24 • 33 = 432
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Next Steps
Read Examples 2-7 Work Problems in text on p. 216
15-20, all; 35-55, odds; 63-67, all Do Online homework corresponding to this
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