Least Common Multiple Calculator

LCM Calculation:

\[ \text{LCM}(a, b) = \frac{|a \times b|}{\text{GCD}(a, b)} \]

Least Common Multiple (LCM):

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1. What is Least Common Multiple?

The Least Common Multiple (LCM) of two integers is the smallest positive integer that is divisible by both numbers. It's a fundamental concept in number theory with applications in fractions, scheduling, and cryptography.

2. How Does the Calculator Work?

The calculator uses the relationship between LCM and GCD:

\[ \text{LCM}(a, b) = \frac{|a \times b|}{\text{GCD}(a, b)} \]

Where:

\( a \) — First positive integer
\( b \) — Second positive integer
GCD — Greatest Common Divisor

Explanation: The GCD is calculated using the Euclidean algorithm, then LCM is derived from the product of the numbers divided by their GCD.

3. Importance of LCM Calculation

Details: LCM is essential for solving problems involving fractions (finding common denominators), scheduling repeating events, and in cryptographic algorithms.

4. Using the Calculator

Tips: Enter two positive integers. The calculator will compute their LCM using the most efficient method.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between LCM and GCD?
A: LCM finds the smallest shared multiple, while GCD finds the largest shared divisor of two numbers.

Q2: Can LCM be calculated for more than two numbers?
A: Yes, by iteratively applying LCM(a,b,c) = LCM(LCM(a,b),c).

Q3: What's the LCM of prime numbers?
A: The LCM of two distinct primes is their product. For the same prime, it's the number itself.

Q4: How does LCM relate to fractions?
A: LCM is used to find the least common denominator when adding or subtracting fractions.

Q5: What's the time complexity of this calculation?
A: The Euclidean algorithm for GCD (and thus LCM) has O(log(min(a,b))) time complexity.