About Least Common Multiple (LCM)

The least common multiple (LCM) of two or more numbers is the smallest positive number that is divisible by all of them without a remainder. It's a fundamental concept in mathematics with applications in fractions, ratios, and various real-world scenarios.

How to Calculate LCM

There are several methods to calculate the LCM:

Method 1: Using Prime Factorization

1. Find the prime factorization of each number.
2. Take each prime factor to the highest power it appears in any of the factorizations.
3. Multiply these prime factors together.

Method 2: Using GCD (Greatest Common Divisor)

The LCM can be calculated using the formula:

Properties of LCM

• LCM(a, b) ≥ max(a, b)
• LCM(a, b) = a × b ÷ GCD(a, b)
• If a divides b, then LCM(a, b) = b
• LCM(a, b, c) = LCM(LCM(a, b), c)

Applications of LCM

Field	Application
Mathematics	Adding and subtracting fractions with different denominators
Computer Science	Scheduling algorithms, memory allocation
Engineering	Gear ratios, timing circuits
Everyday Life	Finding when events with different periods will coincide

Example Problems

Example 1: Find the LCM of 8, 12, and 20.

Example 2: Two buses run on the same route. One completes a round trip every 30 minutes, and the other every 45 minutes. If they both start at the same time, after how many minutes will they meet again at the starting point?