LCM Calculator
Find the least common multiple (LCM) of two or more numbers.
Results
What Is the Least Common Multiple?
The Least Common Multiple (LCM) of two or more integers is the smallest positive integer that is divisible by all of them. The LCM of 12 and 18 is 36, because 36 is the smallest number divisible by both 12 and 18.
Method 1: Prime Factorization
Factor each number into primes. Take the highest power of each prime that appears in any factorization, then multiply them together. Example: 12 = 2² × 3, 18 = 2 × 3². LCM = 2² × 3² = 4 × 9 = 36.
Method 2: Using GCF
LCM(a, b) = (a × b) / GCF(a, b). For a=12, b=18: GCF=6, so LCM = (12×18)/6 = 216/6 = 36. This method is fast for two numbers when you know the GCF.
Method 3: Listing Multiples
List multiples of each number until a common one appears. Multiples of 12: 12, 24, 36, 48... Multiples of 18: 18, 36, 54... First common: 36. This method works but is slow for large numbers.
Applications
LCM is used to find the common denominator when adding fractions (1/12 + 1/18 requires LCD = LCM(12,18) = 36), scheduling (event cycles that sync up), and gear systems (when two gears return to the same relative position).
Worked Example: Adding Fractions
Add 5/12 + 7/18. First find LCM(12, 18): 12=2²×3, 18=2×3², LCM=2²×3²=36. Convert: 5/12 = 15/36 and 7/18 = 14/36. Add: 15/36 + 14/36 = 29/36. Since GCF(29, 36) = 1 (29 is prime), this is already in lowest terms.
LCM Reference Table
| Numbers | GCF | LCM | Relationship: a×b = LCM×GCF |
|---|---|---|---|
| 4 and 6 | 2 | 12 | 4×6=24, 12×2=24 ✓ |
| 8 and 12 | 4 | 24 | 8×12=96, 24×4=96 ✓ |
| 7 and 11 | 1 | 77 | 7×11=77, 77×1=77 ✓ |
| 15 and 25 | 5 | 75 | 15×25=375, 75×5=375 ✓ |
| 9 and 6 | 3 | 18 | 9×6=54, 18×3=54 ✓ |
Frequently Asked Questions
LCM(4,6) = 12. Multiples of 4: 4, 8, 12... Multiples of 6: 6, 12... First common: 12.
LCM(a,b) × GCF(a,b) = a × b. So LCM = (a × b) / GCF. For 12 and 18: LCM = (12×18)/6 = 36.
When adding fractions with different denominators (to find the LCD), when scheduling events that repeat at different intervals, and in problems involving cycles.
The LCM of two different prime numbers p and q is simply p × q, since they share no common factors greater than 1.
LCM(a, a) = a. A number's LCM with itself is just the number.
Apply LCM iteratively: LCM(a, b, c) = LCM(LCM(a, b), c). For example, LCM(4, 6, 10): first LCM(4, 6) = 12, then LCM(12, 10) = 60. Alternatively, factor all numbers into primes and take each prime factor to the highest power that appears in any number: 4=2², 6=2×3, 10=2×5 → LCM = 2²×3×5 = 60.
If Bus A comes every 12 minutes and Bus B comes every 18 minutes, and they both depart at noon, LCM(12,18) = 36 tells you the buses will next depart together at 12:36. Similarly, if two machines need maintenance every 8 and 6 days respectively, they will both need maintenance on the same day every LCM(8,6) = 24 days.