Understanding GCD & LCM
These two functions are core tenets of arithmetic, specifically when working with fractions, prime factorization, and planetary orbital alignments.
Greatest Common Divisor (GCD)
Also known as the Highest Common Factor (HCF), the GCD is the largest positive integer that divides exactly into two or more numbers without leaving a remainder. It is primarily used to reduce fractions to their simplest form.
- Example: For the numbers 12 and 18, the numbers that divide evenly into 12 are (1,2,3,4,6,12) and into 18 are (1,2,3,6,9,18). The largest number shared in both lists is 6.
Least Common Multiple (LCM)
The LCM is the smallest integer that is perfectly divisible by both numbers. It is primarily used when adding or subtracting fractions to find a common denominator.
- Example: For 12 and 18, the multiples of 12 are (12,24,36,48...) and 18 are (18,36,54...). The smallest multiple they share is 36.
The Calculation Algorithm
To calculate the GCD in software, we utilize the Euclidean Algorithm. It dictates that the GCD of two numbers also divides their difference. We perform continuous modulo division (a % b) until the remainder is 0. Once the GCD is established, calculating the LCM is easily derived using the formula: LCM = (|a × b|) ÷ GCD(a, b).