LCM Calculator

How to use the LCM calculator

The least common multiple of a set of whole numbers is the smallest number that every one of them divides into without remainder. It is the number you need when adding fractions, when working out when two repeating events coincide, and whenever several different cycle lengths have to line up. Three methods find it and they suit different situations.

Listing multiples is the slowest and the most convincing. Write out 12, 24, 36, 48 and 18, 36, 54, and the first number appearing in both lists is 36. It takes longer than the alternatives but it shows exactly what the answer means, which is why it is worth doing once before switching to something quicker.

The shortcut is faster: multiply the two numbers and divide by their greatest common factor. Twelve times eighteen is 216, and dividing by their greatest common factor of 6 gives 36. This works because the product counts every shared factor twice, and dividing by the greatest common factor removes exactly one copy of each.

The prime factorisation method is the one that scales and explains. Break every number into primes, then take each prime that appears anywhere, raised to the highest power it reaches in any of them. Twelve is 2² × 3 and eighteen is 2 × 3²; the highest powers are 2² and 3², and 4 × 9 is 36.

The job it does most often: common denominators

Adding fractions needs both to sit over the same denominator, and any common multiple of the two denominators will work. Using the least one keeps the numbers small and usually means no reducing afterwards. Multiplying the denominators together always works too and often leaves a fraction that still needs reducing.

The same reasoning covers anything cyclical. Two buses leaving every 12 and 18 minutes depart together every 36 minutes. Two gears with 12 and 18 teeth return to their starting alignment after 36 teeth have passed. The least common multiple is where repeating things resynchronise.

Zero, one, and numbers that share nothing

When two numbers are coprime — sharing no factor above one — their least common multiple is simply their product. Seven and thirteen have nothing in common, so the first number they both divide is 91. That is also the largest the least common multiple can ever be for a given pair.

Zero is the case worth stating. Under the definition taught in schools — the least POSITIVE common multiple — zero has no least common multiple with anything, because every multiple of zero is zero and zero is not positive. Some programming libraries adopt the convention that it returns zero instead. This page follows the school definition and declines the input rather than printing a number that means something different depending on who is reading.

The formula, worked line by line

One shortcut, one general method, and a rule about when the shortcut stops applying. The general method never fails, which is why it is worth knowing even though it is slower to write out.

Seeing it that way also explains the failure with three numbers: the shared factors no longer appear a predictable number of times in the product, so there is no single quantity to divide out.

lcm(a, b) = a × b ÷ gcf(a, b)
lcm(a, b, c) = lcm(lcm(a, b), c)
lcm = product of each prime, raised to its highest power
coprime numbers: lcm(a, b) = a × b
if a divides b, then lcm(a, b) = b

When the LCM is one of the inputs

If one number already divides another, the larger one is the answer. Two divides eight, so the least common multiple of 2 and 8 is 8, not 16. It surprises people who expect the answer to be bigger than everything they entered, and it is the reason the three-number shortcut fails on 2, 4 and 8.

A number worth recognising

That is why 2,520 turns up so often in puzzles and in problems about coincidence — it is the smallest number divisible by everything from one to ten, and adding 9 to the list is what pushes it there rather than the 10.

Questions people ask