LCM Calculator
Least Common Multiple — up to 3 numbers
Leave blank to find LCM of two numbers only
Enter two or three whole numbers to find their Least Common Multiple.
LCM
12
Least Common Multiple
Use as Fraction LCD
Prime Factorization Method
Relationships
LCM
GCF
LCM × GCF
First multiple shared
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Least Common Multiple Methods & Applications
The Least Common Multiple (LCM) is the smallest positive number that is a multiple of all given numbers. LCM is most commonly used to find the Lowest Common Denominator (LCD) when adding or subtracting fractions with different denominators.
Formula: LCM(a, b) = (a × b) / GCF(a, b). This makes GCF and LCM closely related — knowing one gives you the other instantly.
LCM for Fraction Addition
To add 1/4 + 1/6: LCM(4,6) = 12. Convert: 3/12 + 2/12 = 5/12. Using LCM as the LCD keeps numbers as small as possible.
Prime Factorization Method
LCM(12,18): 12 = 2²×3. 18 = 2×3². Take highest power of each prime: 2²×3² = 4×9 = 36. Always take the highest, not the lowest (GCF takes lowest).
Listing Multiples Method
LCM(4,6): Multiples of 4: 4, 8, 12, 16... Multiples of 6: 6, 12, 18... First in common: 12. Simple for small numbers, slow for large ones.
Real-World Uses
Scheduling: two events repeat every 4 and 6 days. They coincide every LCM(4,6) = 12 days. Also used for gear ratios, tiling patterns, and synchronization problems.
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LCM Questions
What is the Least Common Multiple (LCM)?+
The LCM is the smallest positive number that is a multiple of all given numbers. LCM(4, 6) = 12, because 12 is the smallest number that both 4 and 6 divide into evenly (4×3=12, 6×2=12). LCM is also called the Lowest Common Multiple or Smallest Common Multiple. Every common multiple of the numbers is a multiple of the LCM.
How do I find LCM using the GCF formula?+
LCM(a, b) = (a × b) / GCF(a, b). Example: LCM(12, 18). GCF(12, 18) = 6. LCM = (12 × 18) / 6 = 216 / 6 = 36. This is the most efficient method for two numbers because you only need to find the GCF, which is quick with the Euclidean algorithm. For three numbers, find LCM of the first two, then LCM of that result and the third.
Why do we need LCM when adding fractions?+
To add fractions, you need a common denominator. The LCM of the denominators is the LCD (Lowest Common Denominator). Example: 1/4 + 1/6. LCD = LCM(4, 6) = 12. Convert: 3/12 + 2/12 = 5/12. Using the LCM (not just any common multiple) keeps numbers as small as possible, which reduces the simplification needed afterward. If you used 24 instead: 6/24 + 4/24 = 10/24, which needs simplifying to 5/12.
How do I find LCM using prime factorization?+
Factor each number into primes. For each prime that appears in any factorization, take the highest power it appears at. Multiply these together. Example: LCM(12, 18, 20). 12 = 2²×3. 18 = 2×3². 20 = 2²×5. Highest powers: 2², 3², 5¹. LCM = 4×9×5 = 180. Key difference from GCF: LCM uses highest powers, GCF uses lowest powers.
What is LCM of numbers with no common factors?+
If numbers are coprime (GCF = 1), their LCM is simply their product. LCM(7, 11) = 7 × 11 = 77. LCM(4, 9) = 36 (GCF(4,9) = 1, so LCM = 4 × 9 = 36). This makes sense: if they share no factors, the smallest number divisible by both must contain all factors of each, so it equals their product.
How do I find LCM of three numbers?+
Find LCM of the first two numbers, then find LCM of that result and the third number. Example: LCM(4, 6, 10). LCM(4, 6) = 12. LCM(12, 10): GCF(12,10) = 2. LCM = (12×10)/2 = 60. Answer: LCM(4, 6, 10) = 60. Verify: 60/4=15, 60/6=10, 60/10=6. All divide evenly. This chaining method works for any number of inputs.
What is the relationship between LCM and GCF?+
For two numbers a and b: GCF(a,b) × LCM(a,b) = a × b. This identity is extremely useful: if you find one, you can calculate the other. Example: a=12, b=18. GCF = 6. LCM = (12×18)/6 = 36. Verify: 6×36 = 216 = 12×18. GCF is always ≤ min(a,b) and LCM is always ≥ max(a,b).
What is LCM(a, a) for any number a?+
LCM(a, a) = a. The smallest multiple both a and a share is a itself. Similarly, GCF(a, a) = a. These edge cases are consistent with the formula: LCM(a,a) = (a×a)/GCF(a,a) = a²/a = a. Also: LCM(a, 1) = a, and GCF(a, 1) = 1 for any positive integer a.
How is LCM used in real-life scheduling problems?+
If event A repeats every 4 days and event B every 6 days, they coincide every LCM(4,6) = 12 days. Traffic lights: if one cycle takes 60 seconds and another 90 seconds, they sync every LCM(60,90) = 180 seconds = 3 minutes. Gear teeth: if gears have 12 and 18 teeth, a specific tooth pair meets again after LCM(12,18) = 36 rotations total. LCM solves any synchronization problem.
What is the LCD and how is it different from LCM?+
LCD (Lowest Common Denominator) is just the LCM applied specifically to fraction denominators. They are the same mathematical concept. When adding 1/4 + 1/6, the LCD is LCM(4,6) = 12. The term LCD is used in the context of fractions; LCM is the general mathematical term. Both refer to the smallest number that all given numbers divide into evenly.