Find the Least Common Multiple (LCM) of 2 or more numbers with step-by-step solutions. Multiple calculation methods available.
Separate numbers with commas or spaces
• LCM(12, 15, 75) = 300
• LCM(6, 8) = 24
• LCM(4, 6, 12) = 12
Enter 2 or more numbers and click Calculate
Separate numbers with commas or spaces
The Least Common Multiple (LCM), also known as the Lowest Common Multiple, is the smallest positive integer that is divisible by all the given numbers without leaving a remainder. For example, the LCM of 4 and 6 is 12, because 12 is the smallest number that both 4 and 6 divide into evenly.
LCM is fundamental in mathematics and has practical applications in adding and subtracting fractions, solving problems involving cycles or patterns, scheduling, and music theory. Understanding how to find the LCM helps develop number sense and problem-solving skills essential for algebra and higher mathematics.
Our LCM Calculator provides three different methods to find the Least Common Multiple: Prime Factorization (most efficient), Listing Multiples (most visual), and Formula Method (fastest for 2 numbers). Each method includes complete step-by-step explanations to help you understand the process and learn the concept thoroughly.
This is the most efficient method for finding LCM, especially with larger numbers or multiple numbers.
12 = 2² × 3
18 = 2 × 3²
24 = 2³ × 3
LCM = 2³ × 3² = 8 × 9 = 72
This method is more visual and helps understand what LCM actually means.
Multiples of 4: 4, 8, 12, 16, 20, 24, 28...
Multiples of 6: 6, 12, 18, 24, 30...
First common multiple = LCM = 12
This is the fastest method when you only have two numbers.
LCM(a, b) = (a × b) / GCF(a, b)
Step 1: Find GCF(12, 18) = 6
Step 2: LCM = (12 × 18) / 6
Step 3: LCM = 216 / 6 = 36
Select Prime Factorization, Listing Multiples, or Formula method from the dropdown. Prime Factorization is recommended for most cases.
Type 2-10 positive integers separated by commas or spaces. Example: "12, 15, 75" or "12 15 75".
Click Calculate button (or press Enter) to see the LCM, GCF, prime factorizations, and complete step-by-step solution.
Review the prime factorizations, step-by-step explanation, and GCF to fully understand how the LCM was calculated.
For any two numbers a and b: LCM(a,b) × GCF(a,b) = a × b
This relationship is used in the formula method to quickly calculate LCM from GCF.
To add fractions with different denominators, you need to find the LCM of the denominators to get a common denominator. Example: 1/4 + 1/6 requires LCM(4,6) = 12.
If two events occur at different intervals, LCM tells you when they'll coincide. Example: If bus A comes every 15 minutes and bus B every 20 minutes, they'll arrive together every LCM(15,20) = 60 minutes.
LCM is used to find when different rhythmic patterns align. If one pattern repeats every 3 beats and another every 4 beats, they align every LCM(3,4) = 12 beats.
In mechanical engineering, LCM helps calculate when gears with different tooth counts will return to the same position. Essential for designing gear systems and timing mechanisms.
Finding the smallest quantity that can be evenly divided into different package sizes. Example: Items sold in packs of 4 and 6 - LCM(4,6) = 12 is the smallest order that can be packaged either way.
Calculating when celestial events align, such as when planets return to the same relative positions. LCM of their orbital periods determines synchronization points.
Wrong: LCM(4, 6) = 2 (that's the GCF!)
Correct: LCM(4, 6) = 12 ✓
Remember: LCM is always ≥ the largest number, GCF is always ≤ the smallest number.
Wrong: LCM(4, 6) = 4 × 6 = 24 ✗
Correct: LCM(4, 6) = 12 ✓
Multiplying gives a common multiple, but not necessarily the least one.
Wrong: 12 = 2 × 6, so LCM(12, 18) uses 6 as a factor ✗
Correct: 12 = 2² × 3, fully factor into primes ✓
Always break down completely into prime factors.
Wrong: For 2² and 2³, use 2² in LCM ✗
Correct: For 2² and 2³, use 2³ in LCM ✓
LCM uses highest powers (GCF uses lowest powers).
Wrong: 24 is common to 4 and 6, so that's the LCM ✗
Correct: Check smaller multiples first; LCM(4,6) = 12 ✓
Find the LEAST (smallest) common multiple, not just any common multiple.
If one number is a multiple of the other, the larger number IS the LCM. Example: LCM(6, 12) = 12.
For just 2 numbers, LCM(a,b) = (a × b) / GCF(a,b) is faster than prime factorization.
If numbers share no common factors (co-prime), their LCM is simply their product. Example: LCM(7, 11) = 77.
The LCM of any numbers with a prime number among them will always be a multiple of that prime.
The LCM should be divisible by each of the original numbers. Quickly divide to check your work.
Technically, LCM is undefined for 0 because every number is a divisor of 0, so there's no "least" common multiple. In practice, we only find LCM of positive integers. Our calculator requires positive integers only.
No, the LCM is always greater than or equal to the largest number in the set. It must be a multiple of each number, so it can't be smaller than any of them. The only time LCM equals the largest number is when that number is already a multiple of all the others.
LCD (Least Common Denominator) is the LCM of the denominators of two or more fractions. They're the same mathematical concept - LCD is just LCM applied specifically to fraction denominators. When adding fractions, you find the LCD to get a common denominator.
The LCM of two prime numbers is always their product. Since prime numbers share no common factors (their GCF is 1), you multiply them together. For example, LCM(7, 11) = 77, and LCM(3, 5) = 15.
To add fractions, they must have the same denominator. The LCM of the denominators gives us the smallest common denominator, making the calculation easier and resulting in a simpler answer. For example, to add 1/4 + 1/6, we use LCM(4,6) = 12 as the common denominator.
If all numbers are identical, the LCM is that number itself. For example, LCM(5, 5, 5) = 5. This makes sense because the number is already a multiple of itself and is the smallest such multiple.
LCM is typically defined only for positive integers. While you could extend the concept to negative numbers, the standard mathematical definition and most practical applications use only positive integers. Our calculator works with positive integers only.
Mathematically, you can find the LCM of any quantity of numbers. Our calculator supports 2-10 numbers at once, which covers the vast majority of practical applications. To find LCM of more numbers, you can find it in groups, then find the LCM of those results.
The LCM can be as large as the product of all the numbers (which happens when the numbers are co-prime). For very large numbers or many numbers, the LCM can become extremely large. Our calculator handles numbers up to standard JavaScript integer limits.
For 2 numbers, the formula method (using GCF) is fastest. For 3+ numbers, prime factorization is most efficient. Listing multiples works well for small numbers but becomes tedious with larger numbers. Prime factorization is recommended as the general-purpose method.
Understanding the Least Common Multiple is essential for mastering fractions, solving mathematical problems, and recognizing patterns in everyday situations. Whether you're a student learning to add fractions, a teacher preparing lessons, or someone working on scheduling or engineering problems, finding the LCM is a valuable skill.
Our LCM Calculator provides three different methods—Prime Factorization, Listing Multiples, and Formula Method—each with complete step-by-step explanations. This not only gives you quick answers but helps you understand the underlying mathematics and learn the most efficient approaches for different situations.
Use this calculator to check your homework, learn new methods, solve real-world problems, or simply explore the fascinating patterns in number theory. Remember that practice and understanding go hand-in-hand—the more you work with LCM, the more intuitive it becomes. Happy calculating!