section 5.1 number theory

Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Section 5.1

Number Theory


What You Will Learn

Introduction to Number TheoryPrime NumbersComposite NumbersPrime Factorization

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Number Theory

The study of numbers and their properties.The numbers we use to count are called counting numbers, or natural numbers, denoted by N.

N = {1, 2, 3, 4, 5, …}

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Factors

The natural numbers that are multiplied together are called factors of the product.A natural number may have many factors. The factors of 18 are 1, 2, 3, 6, 9 and 18.

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Divisors

If a and b are natural numbers, we say that a is a divisor of b or a divides b, symbolized a|b if the quotient of b divided by a has a remainder of 0.If a divides b, then b is divisible by a.

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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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Divisibility Rules

265; ends in 5The number ends in 0 or 5.

5

924; 24 is divisible by 4

The number formed by the last two digits of the number is divisible by 4.

4

924; 8 + 4 + 6 = 18 and 18 is divisible by 3

The sum of the digits of the number is divisible by 3.

3

924; evenThe number is even.2

ExampleTestDivisible by

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Divisibility Rules

730; ends in 0The number ends in 0.10

837; 8 + 3 + 7 = 18 and 18 is divisible by 9

The sum of the digits of the number is divisible by 9.

9

5824; 824 is divisible by 8

The number formed by the last three digits of the number is divisible by 8.

8

924; divisible by both 2 and 3

The number is divisible by both 2 and 3.

6

ExampleTestDivisible by

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The Fundamental Theorem of Arithmetic

Every composite number can be expressed 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 FactorizationsMethod 1: Branching

Select any two numbers whose product is the number to be factored.If the factors are not prime numbers, continue factoring each number until all numbers are prime.

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Example 2: Prime Factorization by Branching

Thus, the prime factorization of 1500 = 2 • 2 • 3 • 5 • 5 • 5 = 22 • 3 • 53

Write 1500 as a product of primes.Solution

or

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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.

Method 2: Division

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Example 2: Prime Factorization by Division

1500 = 2 • 2 • 3 • 5 • 5 • 5 = 22 • 3 • 53

Write 1500 as a product of prime numbers.Solution

150075037512525

22355

5

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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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To Determine the GCD of Two or More Numbers1.Determine the prime

factorization of each number.2.List each prime factor with

smallest exponent that appears in each of the prime factorizations.

3.Determine the product of the factors found in step 2.

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Example 4: Using Prime Factorization to Determine the GCDDetermine the GCD of 54 and 90.Solution

54 = 2 • 33

90 = 2 • 32 • 5 Prime factors with smallest exponents that appear in each factorization

2 and 32

The GCD is 2 • 32 = 18.5.1-16


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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To Determine the LCM of Two or More Numbers

1.Determine the prime factorization of each number.

2.List each prime factor with the greatest exponent that appears in any of the prime factorizations.

3.Determine the product of the factors found in step 2.

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Example 6: Using Prime Factorization to Determine the LCMDetermine the LCM of 54 and 90.Solution

54 = 2 • 33

90 = 2 • 32 • 5 Prime factors with greatest exponents that appear in either factorization

2, 33 and 5The LCM is 2 • 33 • 5 = 90.5.1-19


Search for Larger Prime NumbersMore than 2000 years ago, the Greek mathematician Euclid proved that there is no largest prime number.Mathematicians, however, continue to strive to find larger and larger prime numbers.

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Mersenne PrimesMarin Mersenne (1588–1648), a seventeenth-century monk, found that numbers of the form 2n – 1 are often prime numbers when n is a prime number

Numbers of the form 2n – 1 are referred to as Mersenne primes

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Mersenne PrimesThe first 10 Mersenne primes occur whenn = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89

The first time the expression does not generate a prime number, for prime number n, is when n is 11

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Mersenne PrimesLargest prime discovered August 23, 2008 by Edson Smith, UCLA

243,112,609 – 1

It is 12,978,189 digits longWritten in 12-point font it is 17 miles long

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Fermat Numbers

Pierre de Fermat (1601 – 1665) conjectured that each number of the form , now referred to as a Fermat number, was prime for each natural number n.A conjecture is a supposition that has not been proved or disproved.

22n

1

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Fermat NumbersIn 1732, Leonhard Euler proved that for n = 5, 232 + 1 was a composite number, thus disproving Fermat’s conjecture.Since Euler’s time, mathematicians have been able to evaluate only the sixth, seventh, eighth, ninth, tenth, and eleventh Fermat numbers and each of these numbers has been shown to be composite.

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Goldbach’s ConjectureIn 1742, Christian Goldbach conjectured that every even number greater than or equal to 4 can be represented as the sum of two (not necessarily distinct) prime numbers.For example, 4 = 2 + 2, 6 = 3 + 3, 8 = 3 + 5, 10 = 5 + 5, 12 = 5 + 7.It remains unproven to this day.

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Twin Prime ConjectureTwin primes are primes of the form p and p + 2 (3 and 5, 11 and 13).The conjecture states that there are an infinite number of pairs of twin primes.The largest known twin primes are of the form 65,516,468,355 2333,333 ± 1.The were found by the efforts of two research groups: Twin Prime Search and PrimeGrid on August 6, 2009.

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section 5.1 number theory

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