GCD and LCM: meaning, relationship, and example
The greatest common divisor (GCD) is the largest positive whole number that divides both inputs evenly. The least common multiple (LCM) is the smallest positive whole number that both inputs divide into evenly. They are useful for simplifying fractions, matching repeating cycles, and finding common denominators.
Worked example: 12 and 18
The largest shared factor of 12 and 18 is 6. The multiples of 12 are 12, 24, 36, … and multiples of 18 are 18, 36, …, so their least common multiple is 36.
A useful relationship
For nonzero integers, LCM(a, b) = |a × b| ÷ GCD(a, b). The absolute value makes the LCM nonnegative even if an input is negative. This calculator expects whole numbers and rejects zero inputs; it does not calculate GCD or LCM for decimals.
To simplify a fraction, divide its numerator and denominator by their GCD. To find when two repeating schedules with integer intervals coincide, their LCM can identify the first shared interval. These methods assume exact whole-number values.
Tips for accurate results
The greatest common divisor is the largest number that divides all your inputs, which is what you use to simplify fractions. The least common multiple is the smallest number all of them divide into, which is what you need for common denominators and repeating schedules. For 12 and 18, the GCD is 6 and the LCM is 36.
Common mistakes and limits
For two numbers, GCD times LCM equals the product of the numbers, which is a handy check: 6 × 36 = 216 = 12 × 18. The concepts apply to positive whole numbers, so decimals and negatives should be converted or handled separately. Zero has special rules, and the GCD of 0 and any number is that number.