Common Multiple Calculator
Instantaneously compute the least common multiple (LCM), greatest common divisor (GCD), and list of common multiples with step-by-step prime factorizations.
| Property | Formula / Method | Calculated Value |
|---|---|---|
| Least Common Multiple (LCM) | (a × b) / GCD(a, b) | 36 |
| Greatest Common Divisor (GCD) | Euclidean Algorithm | 6 |
| Product of Numbers | a × b | 216 |
| LCM × GCD Identity Check | LCM × GCD = a × b | 36 × 6 = 216 |
| Multiplier for First Number | LCM / a | 3 (12 × 3 = 36) |
| Multiplier for Second Number | LCM / b | 2 (18 × 2 = 36) |
| First Common Multiples | LCM × 1, 2, 3, 4, 5... | 36, 72, 108, 144, 180 |
Authoritative lookup table showing LCM, GCD, and initial common multiples for standard integer pairs.
| Number Pair | LCM | GCD / GCF | First 4 Common Multiples |
|---|---|---|---|
| 2 and 3 | 6 | 1 | 6, 12, 18, 24 |
| 3 and 4 | 12 | 1 | 12, 24, 36, 48 |
| 4 and 6 | 12 | 2 | 12, 24, 36, 48 |
| 6 and 8 | 24 | 2 | 24, 48, 72, 96 |
| 6 and 9 | 18 | 3 | 18, 36, 54, 72 |
| 8 and 12 | 24 | 4 | 24, 48, 72, 96 |
| 9 and 12 | 36 | 3 | 36, 72, 108, 144 |
| 10 and 15 | 30 | 5 | 30, 60, 90, 120 |
| 12 and 15 | 60 | 3 | 60, 120, 180, 240 |
| 12 and 18 | 36 | 6 | 36, 72, 108, 144 |
| 14 and 21 | 42 | 7 | 42, 84, 126, 168 |
| 15 and 20 | 60 | 5 | 60, 120, 180, 240 |
| 16 and 24 | 48 | 8 | 48, 96, 144, 192 |
| 18 and 24 | 72 | 6 | 72, 144, 216, 288 |
| 20 and 25 | 100 | 5 | 100, 200, 300, 400 |
| 20 and 30 | 60 | 10 | 60, 120, 180, 240 |
| 24 and 36 | 72 | 12 | 72, 144, 216, 288 |
| 30 and 45 | 90 | 15 | 90, 180, 270, 360 |
Understanding Common Multiples & The Least Common Multiple
In arithmetic and number theory, a multiple of an integer n is the product of n and any integer. A common multiple is an integer that is an exact multiple of two or more numbers. Because every positive integer has infinitely many multiples, any set of non-zero integers shares an infinite series of common multiples.
The smallest positive integer that is divisible by every number in the set without leaving a remainder is designated as the Least Common Multiple (LCM), historically termed the lowest common multiple. Every subsequent common multiple in the infinite series is simply an integer multiple of the LCM: k × LCM where k = 1, 2, 3, 4....
The 4 Rigorous Methods to Calculate LCM
1. The Listing Multiples Method
Write out the positive multiples of each number in sequential order until the first identical value appears on all lists:
- Multiples of 12: 12, 24, 36, 48, 60, 72, 84, 96, 108...
- Multiples of 18: 18, 36, 54, 72, 90, 108, 126...
- First common value: 36 (LCM = 36).
2. The Prime Factorization Method (Highest Power Rule)
Decompose each number into its prime factors. The LCM is constructed by multiplying each unique prime factor raised to its highest exponent found across all numbers:
12 = 2 × 2 × 3 = 2² × 3¹18 = 2 × 3 × 3 = 2¹ × 3²- Highest power of 2:
2² = 4 - Highest power of 3:
3² = 9 LCM = 2² × 3² = 4 × 9 = 36
3. The Fundamental Formula Method (GCD Relation)
For any two positive integers a and b, the product of their LCM and GCD equals the product of the numbers themselves:
a × b = LCM(a, b) × GCD(a, b)
Rearranging yields the fastest programmatic algorithm using the Euclidean algorithm for GCD:
LCM(a, b) = (a × b) / GCD(a, b)
4. The Ladder / Cake / Division Method
Write the numbers side-by-side in a row and divide iteratively by common prime factors until no two numbers share a factor. The LCM is the product of all outside divisors and remaining row quotients.
Real-World Applications
1. Gear Synchronization & Meshing
When two meshing gears have 12 teeth and 18 teeth, how many tooth contacts occur before the original alignment returns? The gears align perfectly after every LCM(12, 18) = 36 teeth pass. The smaller gear completes 36 / 12 = 3 rotations, while the larger gear completes 36 / 18 = 2 rotations.
2. Transit & Route Scheduling
If Bus Line A departs every 12 minutes and Bus Line B departs every 18 minutes from a central terminal, they will depart concurrently every LCM(12, 18) = 36 minutes.
3. Adding & Subtracting Fractions (LCD)
Finding a common denominator for 5/12 + 7/18 requires computing the Least Common Denominator (LCD), which is identical to the Least Common Multiple of the denominators: LCD = LCM(12, 18) = 36.
Frequently Asked Questions
LCM(a, b) ≥ max(a, b). It equals the larger number only if the larger number is an exact multiple of the smaller number (e.g., LCM(6, 12) = 12).
LCM(a, b, c) = LCM(LCM(a, b), c). Alternatively, use prime factorization: list prime factors for all three numbers and take the highest power of each prime across all three decompositions.
GCD(a, b) = 1), their LCM is simply their direct product: LCM(a, b) = a × b (for example, LCM(7, 9) = 63).