Download Chapter 5 Number Theory Notes

Chapter 5 Number Theory
Section 5.1 - Prime and Composite Numbers
Number Theory
Number Theory is the branch of mathematics devoted to the study of the properties of the natural numbers.
Remember : Natural numbers are also known as counting numbers. N = {1,2,3,…}
Divisibility
A counting number is divisible by another if the operation of dividing the first by the second leaves a remainder of 0.
Formally: the natural number a is divisible by the natural number b if there exists a natural number k such that a = bk.
If b divides a, then we write b|a.
If the natural number a is divisible by the natural number b, then b is a factor (or divisor) of a, and a is a multiple of b.
The number 30 equals 6 · 5; this product is called a factorization of 30.
Example: Finding Factors
Find all the natural number factors of each number.
a) 24
b) 13
Prime and Composite Numbers
A natural number greater than 1 that has only itself and 1 as factors is called a prime number. A natural number greater
than 1 that is not prime is called composite.
The natural number 1 is neither prime nor composite.
Sieve of Eratosthenes
One systematic method for identifying primes is known as the Sieve of Eratosthenes. To construct a sieve, list all the
natural numbers from 2 through some given natural number. The number 2 is prime, but all multiples of it are composite.
Circle the 2 and cross out all other multiples of 2. Continue this process for all primes less than or equal to the square root
of the last number in the list. Circle all remaining numbers that are not crossed out.
Example: Identifying Prime and Composite Numbers
Decide whether each number is prime or composite
a. 97
b. 59,872
c. 697
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Divisibility Tests
Divisibility tests are an aid to determine whether a natural number is divisible by another natural number. Simple tests are
given on the next two slides. There are tests for 7 and 11, but they are more involved.
Divisible by
Test
2
Number ends in 0, 2, 4, 6, or 8.
3
Sum of the digits is divisible by 3.
4
Last two digits form a number divisible by 4.
5
Number ends in 0 or 5.
6
Number is divisible by both 2 and 3.
8
Last three digits form a number divisible by 8.
9
Sum of the digits is divisible by 9.
10
The last digit is 0.
12
Number is divisible by both 3 and 4.
Example: Apply Divisibility Tests
a. Is the number 4,355,211 divisible by 3?
b. Is the number 2,984,094 divisible by 4?
c. Is the number 4,119,806,514 divisible by 9?
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The Fundamental Theorem of Arithmetic
Every natural number can be expressed in one and only one way as a product of primes (if the order of the factors is
disregarded). This unique product of primes is called the prime factorization of the natural number.
Example: Prime Factorization
a. Find the prime factorization of 240.
b. Find the prime factorization of 1320.
Assignment 41 page 184 ##1-51 odd
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5.2
Large Prime Numbers
Primes are the basis for modern cryptography systems, or secret codes. Mathematicians continue to search for larger and
larger primes.
The theory of prime numbers forms the basis of security systems for vast amounts of personal, industrial, and business
data.
No reasonable formula has ever been found that will consistently generate prime numbers, much less generate all primes.
The most useful attempt, however, is named after the French monk Marin Mersenne (1588-1648).
Mersenne Numbers and Mersenne Primes
For n = 1, 2, 3, …, the Mersenne numbers are those generated by the formula
M n  2n  1.
1.
If n is composite, then Mn is composite.
2.
If n is prime, then Mn may be prime or composite.
The prime values of Mn are called Mersenne primes.
Example:
Find the Mersenne number for n = 2, 3, and 5.
Fermat Numbers
Fermat numbers are another attempt at generating prime numbers.
The Fermat numbers are generated by the formula
n
22  1.
The first five Fermat numbers (through n = 4) are prime.
Euler’s and Escott’s Formulas for Finding Primes
Euler’s prime number formula first fails at n = 41:
n2  n  41
Escott’s prime number formula first fails at n = 80:
n2  79n  1601
Assignment 42 page 189 #1-6,11,12,13,14,16,23-27
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5.3 Selected Topics From Number Theory
Perfect Numbers
A natural number is said to be perfect if it is equal to the sum of its proper divisors.
6 is perfect because 6 = 1 + 2 + 3.
8 is not perfect because 8  1 + 2 + 4
Example: Show that 28 is a perfect number.
Deficient and Abundant Numbers
A natural number is deficient if it is greater than the sum of its proper divisors (proper divisors that add up to less than the
number itself). It is abundant if it is less than the sum of its proper divisors (proper divisors that add up to more than the
number itself).
Example: Decide whether the number is perfect, abundant, or deficient.
a)12
b)10
Amicable (Friendly) Numbers
The natural numbers a and b are amicable, or friendly, if the sum of the proper divisors of a is b, and the sum of the
proper divisors of b is a.
Example: The smallest pair of amicable numbers is 220 and 284. Show they are amicable.
Goldbach’s Conjecture (Not Proved)
Every even number greater than 2 can be written as the sum of two prime numbers.
Example: Write each even number as the sum of two primes.
a) 12
b) 40
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Twin Primes
 Twin primes are prime numbers that differ by 2.
o Examples: 3 and 5, 11 and 13
 There are infinitely many pairs of twin primes.
Fermat’s Last Theorem
For any natural number n  3, there are no triples (a, b, c) that satisfy the equation
a n  bn  c n .
When was Fermat’s Last Theorem proven and by whom?
Another Fermat Theorem
Every odd prime can be expressed as the difference of two squares in one and only one way.
Example: Express 7 as the difference of two squares.
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Section 5.4 Greatest Common Factor and Least Common Multiple
The greatest common factor (GCF) of a group of natural numbers is the largest number that is a factor of all of the
numbers in the group.
Finding the Greatest Common Factor (Prime Factors Method)
Step 1 Write the prime factorization of each number.
Step 2 Choose all primes common to all factorizations, with each prime raised to the least exponent that appears.
Step 3 Form the product of all the numbers in Step 2; this product is the greatest common factor.
Example: Find the greatest common factor of 360 and 1350.
The prime factorization:
360  23  32  5
1350  2  33  52
The GCF is 2 · 32 · 5 = 90.
Two numbers with a greatest common factor of 1 are considered relatively prime.
Finding the Greatest Common Factor (Dividing by Prime Factors Method)
Step 1 Write the numbers in a row.
Step 2 Divide each of the numbers by a common prime factor. Try 2, then 3, and so on.
Step 3 Divide the quotients by a common prime factor. Continue until no prime will divide into all the quotients.
Step 4 The product of the primes in steps 2 and 3 is the greatest common factor.
Example: Find the greatest common factor of 12, 18, and 30.
Since there are no common factors in the last row the GCF is 2 · 3 = 6.
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Finding the Greatest Common Factor (Euclidean Algorithm)
To find the greatest common factor of two unequal numbers, divide the larger by the smaller. Note the remainder, and
divide the previous divisor by this remainder. Continue the process until a remainder of 0 is obtained. The greatest
common factor is the last positive remainder in this process.
Example: Use the Euclidean algorithm to find the greatest common factor of 60 and 168.
The GCF is 12
The least common multiple (LCM) of a group of natural numbers is the smallest natural number that is a multiple of all
of the numbers in the group.
Finding the Least Common Multiple (Prime Factors Method)
Step 1 Write the prime factors of each number.
Step 2 Choose all primes belonging to any factorization; with each prime raised to the largest exponent that appears.
Step 3 Form the product of all the numbers in Step 2; this product is the least common multiple.
Example: Find the least common multiple of 360 and 1350.
3
2
The prime factorization: 360  2  3  5
1350  2  33  52
The LCM is 23 · 33 · 52 = 5400
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Finding the Least Common Multiple (Dividing by Prime Factors Method)
Step 1 Write the numbers in a row.
Step 2 Divide each of the numbers by a common prime factor. Try 2, then 3, and so on.
Step 3 Divide the quotients by a common prime factor. When no prime will divide all quotients, but a prime will divide
some of them, divide where possible and bring any nondivisible quotients down. Continue until no prime will divide any
two quotients.
Step 4 The product of the prime divisors in steps 2 and 3 as well as all remaining quotients is the least common multiple.
Example: Find the least common multiple of 12, 18, and 30.
The LCM is 2 · 3 · 2 · 3 · 5 = 180.
Finding the Least Common Multiple (Formula)
The least common multiple of m and n is given by
LCM 
mn
.
greatest common factor of m and n
Example: Find the LCM of 360 and 1350.
The GCF is 90.
LCM 
360 1350 486000

 5400
90
90
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Section 5.5 The Fibonacci Sequence and the Golden Ratio
The Fibonacci Sequence
A famous problem:
A man put a pair of rabbits in a cage. During the first month the rabbits produced no offspring, but each month thereafter
produced one new pair of rabbits. If each new pair thus produced reproduces in the same manner, how many pairs of
rabbits will there be at the end of one year?
The solution of the problem leads to the Fibonacci sequence. Write the first thirteen terms of the sequence:
Recursive Formula for Fibonacci Sequence
If Fn represents the Fibonacci number in the nth position in the sequence, then
F1  1
F2  1
Fn  Fn2  Fn1 , for n  3.
Example: A Pattern of the Fibonacci Numbers
Find the sum of the squares of the first n Fibonacci numbers for n = 1, 2, 3, 4, 5, and examine the pattern. Generalize this
relationship.
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The Golden Ratio
Golden Rectangle
A golden rectangle is one that can be divided into a square and another (smaller) rectangle the same shape as the original.
L L W

W
L
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