Least Common Multiple of 3 Numbers

What the LCM of three numbers is

The least common multiple of three numbers is the smallest positive integer that each of them divides without a remainder. It always exists — the product of the three is already a common multiple — and it is never smaller than the largest of them.

Where the formula comes from

Every integer above 1 is a product of primes, and that decomposition is unique. A number is divisible by a exactly when it contains every prime of a at least as many times as a does. To be divisible by all three at once, a number must therefore contain each prime to the highest power any of the three reaches — and taking no more than that keeps it the smallest such number.

The two-number shortcut LCM(a, b) = a × b ÷ GCD(a, b) has no three-number analogue. For 4, 6 and 10 the product is 240 and the greatest common divisor is 2, yet 240 ÷ 2 = 120 while the real LCM is 60: with three numbers a shared prime gets subtracted more than once.

How to compute it in three steps

1. Factor all three numbers into primes, for example 12 = 2² × 3, 18 = 2 × 3² and 30 = 2 × 3 × 5.
2. For every prime that appears in at least one of them, keep the highest power it reaches: 2², 3² and 5.
3. Multiply those highest powers: 4 × 9 × 5 = 180. That is the least common multiple.

Factorization against pairs, worked on one example

Two routes lead to the same answer, and it is worth seeing them side by side on 12, 18 and 30. The first lays the prime powers out in a grid and reads the maximum of each column:

The second route never factors anything. It leans on associativity — LCM(a, b, c) = LCM(LCM(a, b), c) — and runs the two-number formula twice:

Which one to use is a question of the numbers, not of taste. The grid is faster while the factorizations are easy to see, and it shows why the answer is what it is. The pairwise route stays fast on numbers far too large to factor, because the greatest common divisor is found by the Euclidean algorithm, which only ever divides with remainder.

Common triples and their LCM

Where three numbers at once actually come up

• Adding three fractions — the common denominator of 1/6 + 1/8 + 1/9 is the LCM of 6, 8 and 9, that is 72
• Repeating schedules — three buses leaving every 12, 18 and 30 minutes meet again after 180 minutes
• Gears and belts — three wheels with 12, 18 and 30 teeth return to the starting position after 180 tooth steps
• Tiling and packing — the smallest square that can be filled by 12 cm, 18 cm and 30 cm strips has a side of 180 cm
• Modular arithmetic — the period of three cycles running together is the LCM of their lengths
• School problems — «find the smallest number divisible by 3, 4 and 5» is this calculation word for word

Limits and caveats

• Order does not matter. LCM(3, 4, 5), LCM(5, 3, 4) and LCM(4, 5, 3) are one and the same 60, so this calculator keeps one page per unordered triple.
• A 1 changes nothing — every integer is divisible by 1, so LCM(1, 2, 3) = LCM(2, 3) = 6.
• A repeated number changes nothing either — a prime power already counted is not counted twice: LCM(2, 4, 4) = LCM(2, 4) = 4.
• Pairwise coprime is not the same as coprime as a set. 6, 10 and 15 have no factor shared by all three, yet every pair among them does — and their LCM is 30, not 900.
• The answer grows fast. Three numbers near a billion give an LCM with 27 digits, which is why this page keeps the value as an exact integer and never as a floating-point number.