LCM Calculator

What Is the Least Common Multiple?

The least common multiple (LCM) of two or more integers is the smallest positive integer that is evenly divisible by each of them. For example, the LCM of 4 and 6 is 12, because 12 is the smallest number that both 4 and 6 divide into without a remainder.

How to Calculate the LCM

There are two common methods:

• Prime factorization: Write the prime factorization of each number, then take the highest power of each prime that appears in any factorization and multiply them together.
• Using the GCD: For two numbers, LCM(a, b) = |a × b| ÷ GCD(a, b). For more than two numbers, apply this iteratively.

For example, to find the LCM of 12, 18, and 24: their prime factorizations are 12 = 2² × 3, 18 = 2 × 3², and 24 = 2³ × 3. Take the highest power of each prime: 2³ × 3² = 8 × 9 = 72. So LCM(12, 18, 24) = 72.

Prime Factorization Explained

Prime factorization breaks a number down into a product of prime numbers. For example, 60 = 2² × 3 × 5. Every positive integer greater than 1 has a unique prime factorization (the Fundamental Theorem of Arithmetic). The calculator shows the factorization of each input number so you can see how the LCM is built from the highest powers of each prime.

Real-World Uses of the LCM

• Adding fractions: The LCM of denominators gives the least common denominator, simplifying addition and subtraction.
• Scheduling: If two events repeat every 4 and 6 days, the LCM (12) tells you when they next coincide.
• Gear ratios: The LCM helps determine when meshed gears return to their starting position.
• Number theory: The LCM is fundamental in modular arithmetic and cryptography.

LCM vs GCD

The LCM and GCD (Greatest Common Divisor) are closely related. The GCD is the largest number that divides all given numbers, while the LCM is the smallest number they all divide into. They are connected by the formula: LCM(a, b) × GCD(a, b) = a × b. This calculator shows both values so you can see the relationship.