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Chapter 7: Factorization and the Fundamental Theorem of Arithmetic
Practice HW p. 53 # 14, Additional Web Exercises
In this chapter, we began to examine some basic about primes and their importance.
Definition: A prime number is a positive integer p 2
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Supplemental Section: Factoring Integers
Practice HW Exercises 14 at the end of these notes
In this section, we examine techniques to obtain the prime factorization.
Square Root Test for Prime Factorization
Fact: A number m is prime if it has no prime
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Chapter 19: Primality Testing and Carmichael Numbers
Practice HW p. 127 # 1, 2, 3a, 4
Recall Fermats Little Theorem says that if p is a prime number and a is an integer where
p / a , then

a p 1 1 (mod p )
If we multiply both sides by a, we obtain the
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Chapter 15: Mersenne Primes and Perfect Numbers
Practice HW p. 108 # 1, 2, 3a, 3b
In this section, we examine how to use Mersenne primes to generate perfect numbers. We
start with the following definition.
Definition 15.1: A number n is a perfect number
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Chapter 5: Divisibility and the Greatest Common Divisor
Practice HW p. 34 # 1, 4, Additional Web Exercises
The Greatest Common Divisor of Two Positive Integers
Definition: The greatest common divisor of two positive integers a and b , denoted as
gcd(a,
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Chapter 10: Congruences, Powers, and Eulers Theorem
Practice HW p. 74 # 2, 3a, Additional Web Exercises
In this section, we state and prove Eulers Theorem and look at its benefits.
EulerPhi Function
Given an integer m, the EulerPhi function, denoted b
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Chapter 14: Mersenne Primes
Practice HW p. 99 # 1, 3
We want to consider under what conditions the quantity a n 1 is prime. We establish
these conditions with the following facts.
Fact 1: a n 1 is even if a is odd.
Proof:
Fact 2: ( a 1)  ( a n 1 )
Proo
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Chapter 6: Linear Equations and the Greatest Common Divisor
Practice HW p. 43 # 1, 2, Additional Web Exercises, p. 53 # 1, 2
In this chapter, we look at how to produce gcd(a, b) using a and b .
Definition: Given two positive integers a and b , the quant
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Chapter 9: Congruences, Powers, and Fermats Little Theorem
Practice HW p. 70 # 1, 2, 4, Additional Web Exercises
In this section, we state and prove Fermats Little Theorem and look at its benefits.
Theorem: Fermats Little Theorem. Let p be a prime numbe
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Chapter 2: The Primitive Pythagorean Triples Theorem
Practice HW p. 19 # 1, 2, 6, 7, Additional Web Exercises
The Pythagorean Theorem
In a right triangle (triangle whose largest angle is 90 degrees), the sum of the squares of
the two legs is equal to th
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Chapter 8: Congruences
Practice HW p. 62 # 1, 2, 3, 5, 6, Additional Web Exercises
In this section, we look at the fundamental concept of modular arithmetic, which is used
in a variety of applications in number theory.
Modular Arithmetic
Definition: Giv