The least common multiple is the smallest number that every one of your numbers divides into. Find it by writing each as primes and taking every prime that appears anywhere, raised to the HIGHEST power it reaches. 12 is 2² × 3 and 18 is 2 × 3², so the answer is 2² × 3² = 36 — not 216, which is what multiplying them gives.
The prime table
Write each number in primes, then build a table: one row per prime that appears ANYWHERE, one column per number, and the power that prime reaches in each.
For 12 and 18 the table has rows for 2 and 3. Under 12 they are 2 and 1; under 18, 1 and 2. Take the larger of each pair — 2 and 2 — and the answer is 2² × 3² = 36.
A dash in the table means the number does not contain that prime at all, which is a power of zero. That case is what makes the rule look arbitrary when it is only stated in words: zero never wins a comparison for the highest, so a prime only one number has still goes into the answer at full strength.
Why not just multiply them
Multiplying the numbers always gives A common multiple. It is just usually not the least one, and being wrong by a factor is a real cost.
12 × 18 = 216, which is six times larger than 36. That six is exactly their greatest common factor — the overlap that the product counts twice. Divide it out and you have the least common multiple, which is the identity the whole calculation rests on.
The one case where multiplying IS right is when the numbers share no factor. 7 and 13 have a greatest common factor of 1, nothing overlaps, and 91 is genuinely the least common multiple.
Counting multiples, and its limit
The definition suggests the method: write out the multiples of each and look for the first one they have in common. For 4 and 6 it works instantly — 4, 8, 12 and 6, 12 — and 12 is the answer.
It is also correct and completely impractical past small numbers. This page says how many steps it would have taken on your inputs: for 360 and 1050 you would have counted thirty-five multiples of 1050 before reaching 12600.
Worth knowing rather than dismissing. Counting is what makes the definition concrete, and the prime table is what makes it usable — and seeing the cost of the first is the honest argument for the second.
The mirror of the greatest common factor
These two questions use the same table and opposite instructions, which is the neatest thing about them.
What do they share? Only primes present in EVERY number, each to its LOWEST power. That is the greatest common factor.
What covers them all? Every prime present in ANY number, each to its HIGHEST power. That is the least common multiple.
Multiply the two answers together for a pair and you get the product of the numbers, because every prime has been taken once at its lower power and once at its higher — which is exactly once in each number.
Adding fractions is the real use
Most people arrive here because of fractions. Adding thirds to quarters needs a common denominator, and the least common multiple of 3 and 4 is 12.
Using the product instead works and costs a reduction afterwards. 1/12 + 1/18 over 216 gives 30/216, which then reduces to 5/36 — and on a marked page a reduction that was never necessary looks exactly like a mistake that got caught.
The Fraction Calculator does the whole job and shows this step inside it. This page is for when the denominator itself is the question.
Buses, gears and repeating cycles
Anything that repeats on a fixed interval meets anything else that repeats on a fixed interval at their least common multiple.
Two buses leaving together, one every 12 minutes and one every 18, next leave together after 36 minutes. Two gears with 12 and 18 teeth return to their starting alignment after 36 teeth have passed — three turns of one and two of the other.
The same argument sizes a repeating pattern: a sequence with period 12 laid against one with period 18 does not repeat as a pair until 36. It is why the answer matters in scheduling, animation timing and anything built on cycles.
Sources and methodology
The least common multiple is settled mathematics; these set out the sequence and the conventions.
Method. The multiple is folded pairwise as a ÷ gcd(a, b) × b, dividing before multiplying so no intermediate is larger than it needs to be. Separately every input is factorised and each prime is taken to the highest power it reaches anywhere; the two results must agree. The multiples table is generated only when the numbers are small enough for it to be readable, and the counting cost is stated in every case. That engine is verified on every change against 64 hand-written assertions, including that the answer is divisible by every input and that gcd × lcm equals the product for every pair up to 120 × 120. The count and the per-case breakdown are published on the formula verification page.
Read the guide
The mirror question — what these numbers share rather than what covers them — is the GCF Calculator, and for a pair each answer determines the other. If the reason you are here is adding fractions, the Fraction Calculator does the whole job and shows this step inside it.