LCM Calculator – Find Least Common Multiple Online

Calculate the Least Common Multiple (LCM) of two or more numbers instantly with our free online LCM calculator. Get accurate results with step-by-step explanations.

Calculate LCM of 3 numbers

First Number

Second Number

Examples:

Understanding LCM (Least Common Multiple)

The Least Common Multiple (LCM) of two or more numbers is the smallest positive number that is divisible by all of them. It's the smallest number that appears in the multiplication tables of all the given numbers.

LCM is essential for adding and subtracting fractions with different denominators – you need the LCM to find a common denominator. It's also used in scheduling problems, like finding when repeating events will coincide.

How to Find LCM

Method 1: Using GCD Formula

LCM(a, b) = (a × b) / GCD(a, b)

Find the GCD first, then use this formula. This is the most efficient method for large numbers.

Method 2: Prime Factorization

Find prime factorization of each number
For each prime, take the highest power that appears
Multiply these together

LCM(12, 18): 12=2²×3, 18=2×3²

LCM = 2² × 3² = 4 × 9 = 36

Worked Examples

Example 1: LCM of 12 and 18

Using GCD method:

GCD(12, 18) = 6

LCM = (12 × 18) / 6 = 216 / 6 = 36

Using prime factorization:

12 = 2² × 3

18 = 2 × 3²

LCM = 2² × 3² = 4 × 9 = 36

Example 2: LCM of 8 and 12

GCD(8, 12) = 4

LCM = (8 × 12) / 4 = 96 / 4 = 24

Verify: Multiples of 8: 8, 16, 24, 32...

Multiples of 12: 12, 24, 36...

First common multiple is 24 ✓

Example 3: LCM of Coprime Numbers (7 and 11)

GCD(7, 11) = 1 (they're coprime)

LCM = (7 × 11) / 1 = 77

For coprime numbers, LCM = product of the numbers.

Example 4: LCM of Three Numbers (6, 8, 12)

First: LCM(6, 8)

GCD(6, 8) = 2

LCM(6, 8) = 48 / 2 = 24

Then: LCM(24, 12)

GCD(24, 12) = 12

LCM(24, 12) = 288 / 12 = 24

LCM(6, 8, 12) = 24

Example 5: Real-World Application

Two buses leave a station. Bus A runs every 15 minutes. Bus B runs every 20 minutes. When will they leave together again?

LCM(15, 20) = (15 × 20) / GCD(15, 20)

GCD(15, 20) = 5

LCM = 300 / 5 = 60 minutes

The buses will leave together every hour.

Quick Fact

The relationship LCM(a,b) × GCD(a,b) = a × b is a beautiful mathematical identity. It means that if you know the GCD, you can instantly find the LCM, and vice versa. This relationship holds for any pair of positive integers.

Frequently Asked Questions

What's the difference between LCM and GCD?

GCD (Greatest Common Divisor) is the largest number that divides both numbers. LCM (Least Common Multiple) is the smallest number that both numbers divide. GCD ≤ both numbers ≤ LCM.

Why is LCM useful for fractions?

To add fractions like 1/12 + 1/18, you need a common denominator. The LCM of 12 and 18 is 36, so convert to 3/36 + 2/36 = 5/36. Using LCM gives the smallest (simplest) common denominator.

Can LCM be smaller than the original numbers?

No. The LCM is always at least as large as the largest input number. If one number divides the other (like 4 and 12), the LCM equals the larger number.

What if the numbers are coprime?

If two numbers share no common factors (GCD = 1), their LCM is simply their product. For example, LCM(7, 11) = 77.

How do I find LCM of more than two numbers?

Find LCM of the first two, then find LCM of that result with the third number, and so on. LCM(a, b, c) = LCM(LCM(a, b), c).

What's the LCM of a number with itself?

LCM(n, n) = n. A number is its own smallest multiple that's also divisible by itself.

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