LCM vs GCD: What They Mean and When You Need Them
Two opposite questions about the same pair of numbers. The LCM and GCD Calculator answers both.
Which is which
The greatest common divisor is the largest number that divides into both exactly. For 12 and 18 it is 6.
The lowest common multiple is the smallest number that both divide into exactly. For 12 and 18 it is 36.
The GCD is never larger than the smaller number. The LCM is never smaller than the larger one. If you have them the wrong way round, that check catches it immediately.
When you need each
GCD simplifies. Reducing a fraction means dividing top and bottom by their GCD. Simplifying a ratio means dividing every part by the GCD - the Ratio Simplifier does exactly that. Cutting a rectangle into the largest possible identical squares uses the GCD of the sides.
LCM synchronises. Adding fractions needs a common denominator, and the lowest one is the LCM - which the Fraction Calculator uses. Two buses leaving every 12 and 18 minutes next depart together after 36. Gears with 12 and 18 teeth return to their starting alignment after 36 teeth have passed.
The pattern: GCD for breaking things down, LCM for finding when things coincide.
Finding them
For the GCD, Euclid's algorithm is over two thousand years old and still the method used. Divide the larger by the smaller, keep the remainder, repeat with the smaller and the remainder until the remainder is zero. The last non-zero remainder is the GCD.
For 48 and 18: 48 divided by 18 leaves 12. 18 divided by 12 leaves 6. 12 divided by 6 leaves 0. The GCD is 6. Three steps, no factorising.
For the LCM, the shortcut is worth memorising: multiply the two numbers and divide by their GCD. For 12 and 18, that is 216 divided by 6, which is 36. Finding the GCD first makes the LCM nearly free.
Two useful facts
If the GCD is 1, the numbers are coprime - they share no factors. Their LCM is then simply their product, and a fraction with coprime top and bottom is already in lowest terms.
For three or more numbers, work pairwise: the GCD of the first two, then of that result and the third. The same for the LCM. The shortcut formula does not extend directly to three numbers, so do it stepwise.
Common questions
Is GCD the same as HCF?
Yes - highest common factor is the British name for the same thing.
What is the GCD of a number and zero?
The number itself, since everything divides zero.
Does the prime factorisation method work?
Yes, and it is slower for large numbers. Take the lowest power of each shared prime for the GCD and the highest power of every prime for the LCM. The Prime Number Checker covers why primes underpin this.
Which do I need for a recurring schedule?
LCM. Anything asking "when do these next coincide" is an LCM question.
Got two numbers to work on?