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Common Multiple Calculator
Find the LCM, GCD, and first N common multiples of two or more numbers. Includes prime factorization table, multiples overlap visualization, and preset number sets.
Common Multiple Calculator
LCM⧉
60
Least Common Multiple of {4, 6, 10} — the smallest positive integer divisible by all
GCD⧉
2
Greatest Common Divisor — largest integer that divides all the numbers
LCM × GCD⧉
120
Product of LCM and GCD (equals a×b only for two numbers)
Numbers Count⧉
3
Computing common multiples for 3 numbers
Common Multiples in Range⧉
8
8 common multiples up to 500
LCM / GCD Ratio⧉
30.00
Higher ratio means numbers share fewer common factors
Multiples Overlap Visualization
Multiples of 4 — 125 total, 8 common
Multiples of 6 — 83 total, 8 common
Multiples of 10 — 50 total, 8 common
■ Unique multiples ■ Common multiples
Prime Factorization
| Number | 2 | 3 | 5 | Factorization |
|---|---|---|---|---|
| 4 | 2 | – | – | 2^2 |
| 6 | 1 | 1 | – | 2 × 3 |
| 10 | 1 | – | 1 | 2 × 5 |
| LCM | 2 | 1 | 1 | 60 |
First 10 Common Multiples
| # | Multiple | = LCM × n |
|---|---|---|
| 1 | 60 | 60 × 1 |
| 2 | 120 | 60 × 2 |
| 3 | 180 | 60 × 3 |
| 4 | 240 | 60 × 4 |
| 5 | 300 | 60 × 5 |
| 6 | 360 | 60 × 6 |
| 7 | 420 | 60 × 7 |
| 8 | 480 | 60 × 8 |
| 9 | 540 | 60 × 9 |
| 10 | 600 | 60 × 10 |
Planning notes, formulas, and examples
About the Common Multiple Calculator
The common multiple calculator finds every shared multiple of two or more whole numbers, starting from the least common multiple (LCM). A common multiple is any number that is divisible by all numbers in the set — for example, 12 is a common multiple of 3 and 4 because 12 ÷ 3 = 4 and 12 ÷ 4 = 3. The LCM is the smallest such value and is the building block for all other common multiples: every common multiple is simply LCM × 1, LCM × 2, LCM × 3, and so on. This calculator also computes the GCD (greatest common divisor), displays full prime factorizations in a comparison table, and generates a visual overlap bar showing how individual multiples relate to shared ones. Finding common multiples is essential for adding fractions with different denominators, scheduling recurring events that align (like bus routes), and synchronizing periodic signals in engineering. Enter up to ten numbers, choose how many multiples to display, and explore the factorization breakdown to see why the LCM takes the highest powers of each prime.
When This Page Helps
Finding the LCM of two numbers is straightforward, but computing it for three or more numbers with full prime factorization and listing many common multiples is time-consuming by hand. This calculator handles any number of inputs, shows the complete prime factorization comparison table, generates customizable lists of common multiples, and visualizes how individual multiples overlap. It is essential for fraction addition (finding common denominators), scheduling problems, and number theory exercises.
How to Use the Inputs
- Enter two or more positive integers separated by commas in the Numbers field (e.g., "4, 6, 10").
- Set the Count field to choose how many common multiples to list (up to 50).
- Set the Max Value to limit the range for the individual multiples display.
- Click a preset like "4, 6, 10" or "2, 3, 5, 7" to load common examples.
- Review the LCM, GCD, and first N common multiples in the output cards.
- Examine the prime factorization table to see how the LCM is computed from highest prime powers.
- Toggle Sort Order to view multiples in ascending or descending order.
Formula used
LCM(a, b) = |a × b| / GCD(a, b)
GCD via Euclidean algorithm: GCD(a, b) = GCD(b, a mod b)
Common multiples: LCM×1, LCM×2, LCM×3, …Example Calculation
Result: LCM = 60, GCD = 2
For 4, 6, 10: GCD = 2, LCM = 60. First 5 common multiples: 60, 120, 180, 240, 300.
Tips & Best Practices
- Check that all inputs use the same scale and assumptions before trusting the result.
- Compare the answer with the worked example or a rough estimate to catch entry mistakes.
How the LCM Is Computed
The LCM of two or more numbers is found by taking the highest power of every prime factor that appears in any of the numbers. For 4 = 2², 6 = 2¹ × 3¹, and 10 = 2¹ × 5¹, the LCM takes 2² (from 4), 3¹ (from 6), and 5¹ (from 10), giving 4 × 3 × 5 = 60. The GCD, conversely, takes the lowest power of each shared prime: just 2¹ = 2. The relationship LCM(a,b) × GCD(a,b) = a × b holds for two numbers but not directly for three or more.
Practical Uses of Common Multiples
Adding fractions requires a common denominator, which is a common multiple of the individual denominators. Using the LCM as the common denominator keeps the numbers as small as possible. In scheduling, if buses arrive every 12 minutes and 15 minutes, they coincide every LCM(12,15) = 60 minutes. Gear systems synchronize when the number of teeth are in LCM relationships, and musical rhythms align at LCM-based intervals (polyrhythms).
The Euclidean Algorithm
The fastest method for computing the GCD is the Euclidean algorithm: repeatedly replace the larger number with the remainder of dividing the two. For example, GCD(48, 18): 48 mod 18 = 12, 18 mod 12 = 6, 12 mod 6 = 0, so GCD = 6. The LCM then follows from LCM = |a × b| / GCD. This algorithm runs in O(log min(a,b)) steps and has been known since ∼300 BC, making it one of the oldest algorithms still in daily use.
Sources & Methodology
Last updated:
Frequently Asked Questions
The LCM is the smallest positive integer that is evenly divisible by every number in the set. All other common multiples are integer multiples of the LCM.
Compute the LCM of the first two numbers, then find the LCM of that result with the third number, and continue for each additional number.
For two numbers a and b: LCM(a,b) × GCD(a,b) = a × b. This identity does not extend directly to three or more numbers.
The LCM is found by taking the highest power of each prime factor that appears in any number. The table shows these exponents clearly.
Yes — if one number divides the other, the LCM equals the larger number. For example, LCM(3, 6) = 6.
Adding fractions (finding a common denominator), scheduling events that repeat at different intervals, gear ratios, and signal synchronization.
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