LCM Calculator

The Least Common Multiple (LCM) of two or more numbers is the smallest positive integer that is evenly divisible by all the given numbers. For example, the LCM of 4 and 6 is 12, because 12 is the smallest number that both 4 and 6 divide into evenly.

The most efficient method uses the Greatest Common Divisor (GCD): LCM(a, b) = (a × b) / GCD(a, b). For example, to find LCM(12, 18): First find GCD(12, 18) = 6, then LCM(12, 18) = (12 × 18) / 6 = 216 / 6 = 36.

For multiple numbers, use the associative property: LCM(a, b, c) = LCM(LCM(a, b), c). Find the LCM of the first two numbers, then find the LCM of that result with the third number, and so on.

LCM (Least Common Multiple) is the smallest number that is a multiple of all given numbers, while GCD (Greatest Common Divisor) is the largest number that divides all given numbers evenly. They are related by the formula: LCM(a, b) × GCD(a, b) = a × b.

LCM is typically defined for positive integers only. For negative numbers, you would first find the LCM of their absolute values. However, this calculator focuses on positive integers for clarity and practical applications.

LCM is used in scheduling (finding when events repeat), music theory (finding common time signatures), engineering (synchronizing gears or cycles), and computer science (finding optimal buffer sizes or timing intervals).

Prime factorization can be used to find LCM by taking each prime factor to its highest power across all numbers. For example, for 12 (2²×3) and 18 (2×3²), the LCM is 2²×3² = 36, using the highest power of each prime factor.

LCM is essential for adding and subtracting fractions with different denominators. The LCM of the denominators becomes the common denominator. For example, to add 1/4 + 1/6, you need LCM(4, 6) = 12 as the common denominator.

No, the LCM is always greater than or equal to the largest number in the set. The LCM equals the largest number only when all other numbers are factors of the largest number.

To verify, check that the number is divisible by all the original numbers and that no smaller positive number has this property. You can also use the relationship: LCM(a, b) × GCD(a, b) = a × b to verify your result.