5.1 Divisibility and Prime Numbers

Number theory forms the backbone of cryptography. Divisibility and prime numbers are fundamental concepts that unlock the secrets of integers. These ideas lay the groundwork for understanding how numbers interact and form the basis for secure communication systems.

Prime factorization, greatest common divisors, and least common multiples build on these basics. These tools allow us to manipulate numbers in powerful ways, enabling the creation of encryption algorithms that keep our digital world safe and secure.

Divisibility and Number Types

Fundamental Concepts of Divisibility and Prime Numbers

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• Divisibility occurs when one integer divides another without a remainder
  • Expressed as "a divides b" or "b is divisible by a"
  • Mathematically written as $a | b$ if there exists an integer $k$ such that $b = ak$
• Prime numbers possess only two factors: 1 and themselves
  • Examples include 2, 3, 5, 7, 11, 13, 17, 19
  • 2 stands as the only even prime number
• Composite numbers have more than two factors
  • Can be expressed as the product of two smaller positive integers
  • Numbers like 4, 6, 8, 9, 10, 12 fall into this category
• Fundamental Theorem of Arithmetic states every positive integer greater than 1 can be uniquely represented as a product of prime numbers
  • Also known as the unique factorization theorem
  • Allows for the expression of any number as a product of prime factors

Prime Factorization and Its Applications

• Prime factorization involves breaking down a number into its prime factors
  • Process of finding the set of prime numbers that multiply together to form the original number
  • Useful in various mathematical operations and problem-solving
• Steps to perform prime factorization:
  1. Start dividing the number by the smallest prime number that divides it evenly
  2. Continue dividing the quotient by prime numbers until the quotient becomes 1
  • 60 can be factored as $2 * 2 * 3 * 5$ or $2^2 * 3 * 5$
• Applications of prime factorization:
  • Simplifying fractions
  • Finding least common multiples and greatest common divisors
  • Solving certain types of Diophantine equations

Greatest Common Divisor and Least Common Multiple

Understanding and Calculating GCD

• Greatest Common Divisor (GCD) represents the largest positive integer that divides two or more integers without a remainder
  • Also referred to as the Greatest Common Factor (GCF) or Highest Common Factor (HCF)
  • For integers a and b, denoted as $GCD(a,b)$ or $\gcd(a,b)$
• Methods to find GCD:
  1. Listing factors: List all factors of each number and identify the greatest common one
  2. Prime factorization: Factor each number and multiply the common prime factors
  3. Euclidean algorithm: An efficient method for larger numbers
• Euclidean algorithm steps:
  1. Divide the larger number by the smaller one
  2. Replace the larger number with the smaller number and the smaller number with the remainder
  3. Repeat until the remainder is zero
  • The last non-zero remainder is the GCD
• Properties of GCD:
  • $\gcd(a,b) = \gcd(b,a)$
  • $\gcd(a,0) = |a|$ for any non-zero a
  • $\gcd(a,1) = 1$ for any integer a

Least Common Multiple and Its Calculation

• Least Common Multiple (LCM) is the smallest positive integer that is divisible by two or more given integers
  • Denoted as $LCM(a,b)$ or $lcm(a,b)$ for integers a and b
• Methods to find LCM:
  1. Listing multiples: List multiples of each number and identify the smallest common one
  2. Prime factorization: Factor each number and multiply all prime factors using the highest power in which they occur
  3. Using GCD: $LCM(a,b) = \frac{|a * b|}{GCD(a,b)}$
• Relationship between LCM and GCD:
  • The product of LCM and GCD of two numbers equals the product of the numbers themselves
  • Expressed as $LCM(a,b) * GCD(a,b) = |a * b|$
• Applications of LCM:
  • Finding common denominators in fraction addition and subtraction
  • Solving scheduling and synchronization problems
  • Determining when periodic events will coincide

Prime Number Generation

Sieve of Eratosthenes and Prime Number Properties

• Sieve of Eratosthenes serves as an ancient and efficient algorithm for finding all prime numbers up to a given limit
  • Named after the ancient Greek mathematician Eratosthenes of Cyrene
• Steps of the Sieve of Eratosthenes:
  1. Create a list of consecutive integers from 2 to the desired limit
  2. Start with the first number in the list (2) and mark all its multiples as composite
  3. Move to the next unmarked number and repeat step 2
  4. Continue until reaching the square root of the limit
• Efficiency of the sieve:
  • Time complexity: $O(n \log \log n)$
  • Space complexity: $O(n)$
• Properties of prime numbers:
  • Infinitude of primes: There are infinitely many prime numbers
  • Distribution of primes becomes less dense as numbers increase
  • Prime Number Theorem describes the asymptotic distribution of prime numbers
• Applications of prime numbers:
  • Cryptography and secure communication (RSA algorithm)
  • Hash functions and data structures
  • Random number generation