Least Common Multiple Calculator: LCM with Steps
Free least common multiple calculator: find LCM of two or more numbers. LCM by prime factorization, LCM of 3 numbers, brute force multiples, GCF formula. Lowest common denominator solver.
What This Calculator Does & Who It's For
Calculator Purpose & Ideal Users
- What You'll Get:LCM result: The smallest positive integer that is a multiple of all your numbers. Show your work: Choose prime factorization (each number as primes, LCM = max exponent per prime), brute force (multiples until first common), or GCF formula (LCM(a,b) = (aรb)/GCF(a,b)). Validation: Positive integers only; comma-separated.
- Ideal Users & Keywords:High volume: least common multiple calculator, LCM calculator, lowest common multiple. Long tail: how to find LCM by prime factorization, LCM of 3 numbers calculator, least common multiple steps, lowest common denominator solver, LCM with steps. Students, teachers, and anyone adding fractions or solving alignment problems. Free online, no sign-up.
- Scope & Limits:Positive integers only (no zero; LCM of 0 is undefined). Up to JavaScript safe integer range. No decimals or fractions.
This least common multiple calculator finds the LCM of two or more positive integers and shows your work using prime factorization, brute force (multiples list), or the GCF formula.
What Is the Least Common Multiple (LCM)?
The least common multiple (LCM) of two or more positive integers is the smallest positive integer that is a multiple of each of them. For example, LCM(4, 6) = 12 because 12 is the smallest number that is a multiple of both 4 and 6. A multiple of a number n is n ร 1, n ร 2, n ร 3, โฆ; a common multiple of two numbers is a number that is a multiple of both. The LCM is used as the lowest common denominator when adding fractions with different denominators and in scheduling and alignment problems. For "What is the LCM of 0?" and "Is LCM the same as LCD?", see the FAQ above.
How to Find LCM by Prime Factorization
To find the LCM using prime factorization, write each number as a product of primes (e.g. 12 = 2ยฒ ร 3, 18 = 2 ร 3ยฒ). The LCM is the product of each prime raised to the maximum exponent it appears with in any of the numbers. For 12 and 18: 2 has max exponent 2, 3 has max exponent 2, so LCM = 2ยฒ ร 3ยฒ = 36. For three or more numbers, include every prime that appears in any number and use the max exponent for each. Select "Prime factorization" in the calculator to see the least common multiple steps. For a quick summary, see the FAQ "How do you find LCM by prime factorization?" above.
Brute Force: Multiples Until First Common
The brute force method lists multiples of each number until the first value that appears in every listโthat value is the LCM. For 4 and 6: multiples of 4 are 4, 8, 12, โฆ; multiples of 6 are 6, 12, โฆ; the first common multiple is 12, so LCM(4, 6) = 12. This method is easy to visualize but can be slow for large numbers. Select "Brute force (multiples)" in the calculator to see the lists.
GCF Formula: LCM(a, b) = (a ร b) / GCF(a, b)
For two numbers, LCM(a, b) = (a ร b) / GCF(a, b). So if you know the greatest common factor, you can get the LCM quickly. For three or more numbers, apply the formula pairwise: LCM(a, b, c) = LCM(LCM(a, b), c). Select "GCF formula" in the calculator to see the LCM of 3 numbers (or more) computed step by step.
Least Common Multiple Calculator FAQ
? What is the LCM of 0?
LCM is usually defined for positive integers only. If one number is 0, then every integer is a multiple of 0, so there is no smallest common multipleโLCM(0, b) is undefined. This calculator accepts only positive integers (no zero) so the LCM is always well-defined.
? How is LCM used in real life?
The least common multiple appears when events repeat on different cycles and you need when they align: repeating tasks (e.g. every 4 days and every 6 days โ every 12 days), gear teeth, adding fractions with different denominators (LCM of denominators = common denominator), and scheduling. It is also used in cryptography and number theory.
? Is LCM the same as LCD?
Not exactly. LCM is the least common multiple of two or more integers. LCD (lowest common denominator) is the LCM of the denominators of fractions. So when you add 1/4 and 1/6, the LCD is LCM(4, 6) = 12. In that sense, the LCD is the LCM of the denominators; this calculator finds LCM, which you can use as the LCD when working with fractions.
? How do you find LCM by prime factorization?
Write each number as a product of primes (e.g. 12 = 2ยฒ ร 3, 18 = 2 ร 3ยฒ). The LCM is the product of each prime raised to the maximum exponent it has in any of the numbers. For 12 and 18: 2 has max exponent 2, 3 has max exponent 2, so LCM = 2ยฒ ร 3ยฒ = 36. Select "Prime factorization" in the calculator to see this breakdown. The article below has the full procedure for three or more numbers.
? Can you find LCM of three or more numbers?
Yes. Enter all numbers separated by commas (e.g. 12, 18, 24). The calculator uses LCM(a, b, c) = LCM(LCM(a, b), c): it finds the LCM of the first two, then the LCM of that result with the next number, and so on. All three methods (prime factorization, brute force, GCF formula) work for any number of positive integers.