LCM Calculator

What is LCM?

The Least Common Multiple (LCM), also called the Lowest Common Multiple, is the smallest positive integer that is divisible by each of the given numbers. It is the smallest number that is a multiple of all the given numbers.

Methods to Find LCM

1. Using GCD Formula

The most efficient method for two numbers:

LCM(a, b) = (a * b) / GCD(a, b)

Example: LCM(12, 18)
GCD(12, 18) = 6
LCM = (12 * 18) / 6 = 216 / 6 = 36

2. Prime Factorization

Find prime factors and multiply the highest powers of each prime:

12 = 2^2 * 3
18 = 2 * 3^2

LCM = 2^2 * 3^2 = 4 * 9 = 36

3. Listing Multiples

List multiples of each number until you find the first common one:

Multiples of 12: 12, 24, 36, 48, ...
Multiples of 18: 18, 36, 54, ...
First common multiple: 36
LCM = 36

Properties of LCM

• Commutative: LCM(a, b) = LCM(b, a)
• Associative: LCM(a, LCM(b, c)) = LCM(LCM(a, b), c)
• Divisibility: LCM(a, b) is always divisible by both a and b
• Minimum: LCM(a, b) >= max(a, b)
• Relationship with GCD: LCM(a, b) * GCD(a, b) = a * b

LCM for Multiple Numbers

To find LCM of more than two numbers:

LCM(a, b, c) = LCM(LCM(a, b), c)

Example: LCM(4, 6, 8) = LCM(LCM(4, 6), 8) = LCM(12, 8) = 24

Applications of LCM

Adding Fractions

LCM is used to find the common denominator when adding or subtracting fractions.

1/4 + 1/6 = ?
LCM(4, 6) = 12
= 3/12 + 2/12 = 5/12

Scheduling Problems

Finding when events that occur at different intervals will coincide.

Bus A comes every 12 minutes
Bus B comes every 18 minutes
Both will arrive together every LCM(12, 18) = 36 minutes

Gear Ratios

Calculating when gears with different teeth will return to starting positions.

Tile and Pattern Problems

Finding the smallest square that can be tiled with rectangular tiles.