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.
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 tool 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.
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.
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, …
Result: LCM = 60, GCD = 2
For 4, 6, 10: GCD = 2, LCM = 60. First 5 common multiples: 60, 120, 180, 240, 300.
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.
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 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.
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. Use this as a practical reminder before finalizing the result.
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. Keep this note short and outcome-focused for reuse.