4.3 Greatest Common Divisor and Least Common Multiple

Numbers are the building blocks of math, and understanding how they relate is crucial. Greatest Common Divisor (GCD) and Least Common Multiple (LCM) are key concepts that help us find connections between numbers and solve complex problems.

These ideas are super useful in real life too. From simplifying fractions to scheduling events, GCD and LCM pop up everywhere. We'll learn different ways to calculate them and see how they're used in cool stuff like cryptography.

Greatest Common Divisor (GCD)

Understanding GCD and Its Properties

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• Greatest Common Divisor represents the largest positive integer that divides two or more numbers without a remainder
• GCD of two numbers a and b denoted as GCD(a,b) or (a,b)
• GCD remains unchanged when replacing either number with the absolute difference between them
• Commutative property applies to GCD calculations GCD(a,b) = GCD(b,a)
• GCD of a number and zero equals the absolute value of that number GCD(a,0) = |a|
• Two numbers are coprime when their GCD equals 1
• Prime numbers are always coprime to any other number except their multiples

Calculating GCD Using Different Methods

• Factor tree method breaks down numbers into their prime factors
  • List prime factors of each number
  • Identify common factors
  • Multiply common factors to find GCD
• Prime factorization method expresses numbers as products of prime factors
  • Write each number as a product of prime factors
  • Identify common prime factors
  • Multiply common prime factors to obtain GCD
• Euclidean algorithm efficiently calculates GCD through repeated division
  • Divide larger number by smaller number
  • Replace larger number with remainder
  • Repeat process until remainder is zero
  • Last non-zero remainder is the GCD
• Long division method involves dividing numbers and using remainders
  • Divide larger number by smaller number
  • Use remainder as new divisor
  • Continue process until remainder is zero
  • Last divisor used is the GCD

Applications and Extensions of GCD

• GCD used in simplifying fractions to lowest terms
• Solving linear Diophantine equations involves GCD
• Extended Euclidean algorithm finds coefficients of Bézout's identity
• Least Common Multiple (LCM) calculation relies on GCD
• Chinese Remainder Theorem utilizes GCD in modular arithmetic
• Public key cryptography employs GCD in RSA algorithm
• Computer algebra systems implement efficient GCD algorithms

Least Common Multiple (LCM)

Understanding LCM and Its Properties

• Least Common Multiple represents the smallest positive integer divisible by two or more numbers
• LCM of two numbers a and b denoted as LCM(a,b) or [a,b]
• LCM always greater than or equal to the larger of the two numbers
• Commutative property applies to LCM calculations LCM(a,b) = LCM(b,a)
• LCM of a number and zero equals zero LCM(a,0) = 0
• Two numbers are coprime if and only if their LCM equals their product
• Prime numbers have LCM equal to their product with any other number

Calculating LCM Using Various Techniques

• Factor tree method breaks down numbers into prime factors
  • List all prime factors of each number
  • Include highest power of each prime factor
  • Multiply all factors to obtain LCM
• Prime factorization method expresses numbers as products of prime factors
  • Write each number as a product of prime factors
  • Take highest power of each prime factor
  • Multiply selected factors to find LCM
• Upward multiplication method involves finding common multiples
  • List multiples of each number
  • Identify first common multiple
  • This common multiple is the LCM
• Division method utilizes GCD to calculate LCM
  • Calculate GCD of the numbers
  • Multiply the numbers
  • Divide product by GCD to obtain LCM

Relationships and Applications of LCM

• GCD and LCM relationship expressed as $GCD(a,b) * LCM(a,b) = a * b$
• LCM used in adding and subtracting fractions with different denominators
• Finding common time intervals in scheduling problems involves LCM
• Synchronization of periodic events utilizes LCM calculations
• Least Common Denominator (LCD) in algebra based on LCM concept
• LCM applied in solving systems of linear congruences
• Cryptography employs LCM in certain encryption algorithms (RSA)
• Computer memory allocation often involves LCM calculations