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GCD & LCM Calculator

How do I find GCD and LCM?

How it is calculated

The greatest common divisor (GCD) is the largest whole number dividing both inputs; the least common multiple (LCM) is the smallest positive number that both divide. The tool finds the GCD with Euclid's algorithm: replace the pair (a, b) with (b, a mod b) until the remainder is zero, and the last non-zero value is the GCD. The LCM then follows from LCM = a ÷ GCD × b, which avoids overflow. Both inputs must be positive whole numbers; decimals are truncated to integers and zero or negatives give no result. The GCD is used to simplify fractions, while the LCM gives a common denominator.

Worked example

Take 12 and 18. Euclid: 18 mod 12 = 6, so (12, 18) → (18, 12) → (12, 6); then 12 mod 6 = 0, so the GCD is 6. LCM = 12 / 6 × 18 = 2 × 18 = 36. So GCD = 6 and LCM = 36. Check: 6 × 36 = 216 = 12 × 18, because for any two numbers GCD × LCM equals their product.

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Common mistakes

  • Mixing up the two: the GCD is never larger than the smaller number, the LCM never smaller than the larger one.
  • Entering decimals such as 12.9, which are cut to 12.
  • Multiplying the numbers (12 × 18 = 216) and calling it the LCM, which only holds when they share no common factor.
  • Using zero or negative numbers, which the tool does not accept.

About this calculator

This calculator finds the greatest common divisor (GCD) — the largest number that divides both inputs evenly — and the least common multiple (LCM) — the smallest number that both inputs divide into evenly — for two integers, using the Euclidean algorithm under the hood.

GCD is used to reduce a fraction to its simplest form (dividing numerator and denominator by their GCD) and comes up in scheduling problems, like finding the largest even group size two teams can both be split into. LCM is used to find a common denominator when adding or subtracting fractions, and to solve 'when do these two repeating events line up again' problems, like two buses that pass a stop every 12 and 18 minutes. Students working through math homework and anyone solving a scheduling or fraction problem by hand use this to check their work quickly.

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