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MasterMath

LCM Calculator

The lowest common multiple of two or three numbers: the smallest number all of them divide into. It is what gives fractions a common denominator.

Lowest common multiple

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Lowest common multiple—
Greatest common divisor—
Factorisations—
Check: GCD × LCM—

How this was worked out

    The rule

    LCM = product of every prime present, each with its highest exponent

    Why it works

    The lowest common multiple is the smallest number that all of them divide exactly. From the prime factorisations it is straightforward: take every prime that appears anywhere, each raised to the largest exponent it reaches.

    How to do it by hand

    1. Factor every number into primes
    2. Take every prime that appears in any of them
    3. Give each the largest exponent it has anywhere
    4. Multiply those together

    What is worth knowing

    The LCM is what you are really finding when you look for a common denominator to add fractions, which is why it turns up in school long before its name does. It is also the answer to any 'when do these coincide again?' question: two buses leaving every 12 and 18 minutes meet again after 36. For two numbers there is a shortcut worth knowing — the LCM is their product divided by their GCD — which avoids factoring entirely when the GCD is easy to find.

    Frequently asked questions

    What is the lowest common multiple?

    The smallest number that all the given numbers divide into exactly.

    How does it relate to the GCD?

    For two numbers, LCM = product ÷ GCD. That shortcut avoids factoring when the GCD is easy.

    Why do I need it for fractions?

    Because adding fractions needs a common denominator, and the lowest common multiple gives the smallest one.

    What if the numbers are coprime?

    Then their LCM is simply their product, since they share no factors to overlap.