GCD and LCM Calculator

The greatest common divisor (GCD) and the least common multiple (LCM) are two core number-theory tools for working with integers. You’ll use the GCD when you want the biggest “shared factor” (for example, to reduce fractions), and the LCM when you want the smallest “shared multiple” (for example, to align repeating schedules or find a common denominator). This calculator accepts a comma-separated list of integers such as 12, 18, 30 and returns both the overall GCD and the overall LCM.

What this calculator computes

• GCD of many numbers: the largest positive integer that divides every input number.
• LCM of many numbers: the smallest positive integer that is a multiple of every input number (when defined).

Key formulas

For two nonzero integers a and b, the GCD is commonly computed using the Euclidean algorithm, and the LCM is linked to the GCD by a simple identity:

Relationship between GCD and LCM (two numbers):

Extending to multiple numbers: the calculator reduces a list step-by-step (associatively):

• gcd(a, b, c) = gcd(gcd(a, b), c)
• lcm(a, b, c) = lcm(lcm(a, b), c)

How to interpret the results

• If the GCD is 1, the numbers are coprime as a set (they share no common factor greater than 1). This is a common check in simplifying ratios or confirming relative primality.
• The LCM is the smallest “alignment point” where all numbers divide evenly. It’s useful for common denominators (fractions) and repeating-cycle problems (timers, schedules, patterns).
• For inputs that include negative numbers, GCD/LCM are typically reported as positive values based on absolute magnitude.

Worked example (multiple numbers)

Input: 12, 18, 30

Step 1 (GCD):

• gcd(12, 18) = 6
• gcd(6, 30) = 6

Result: GCD = 6

Step 2 (LCM):

• lcm(12, 18) = |12×18| / gcd(12,18) = 216 / 6 = 36
• lcm(36, 30) = |36×30| / gcd(36,30)
• gcd(36, 30) = 6, so lcm = 1080 / 6 = 180

Result: LCM = 180

GCD vs LCM: quick comparison

Limitations and assumptions

• Integers only: Inputs should be whole numbers. Decimals or fractions should be converted to integers first (e.g., scale by a common factor).
• Zero handling: gcd(0, n) = |n|. For LCM, many definitions set lcm(0, n) = 0; if any input is 0, the overall LCM is typically reported as 0.
• Sign handling: Results are usually returned as positive values using absolute values.
• Large outputs: LCM can grow very quickly for multiple inputs; extremely large integers may overflow some JavaScript number handling unless big-integer arithmetic is used.

More examples to try

• 8, 20 → GCD 4, LCM 40
• 7, 9, 28 → GCD 1, LCM 252
• -6, 15 → GCD 3, LCM 30
• 0, 12, 18 → GCD 6, LCM 0 (by common convention)