Least Common Multiple Calculator - LCM of Two or More Numbers

The least common multiple (LCM) of a set of positive integers is the smallest positive integer that every member of the set divides. It is the first time the multiplication tables of those numbers line up. If two gears have 12 and 18 teeth, they return to the same mesh after 36 teeth; if a third gear with 30 teeth joins the train, the first common turn is 180 teeth. That is why the default example on this page is 12, 18, 30 → LCM 180.

This free online LCM calculator is search-focused: you paste two or more whole numbers and you get one LCM, computed with JavaScript BigInt so the intermediates stay exact. The identity is lcm(a, b) = |a × b| / gcd(a, b), folded across the list. The GCD of the same set is shown as an optional check - it is not the headline. If you wanted a combined GCD + LCM + common-factor dump, that is the common factor calculator. If you wanted only the greatest shared factor, that is the greatest common factor calculator.

Display is capped at 8000 digits. An LCM can grow much faster than the inputs (coprime 500-digit numbers have a ~1000-digit LCM; many pairwise-coprime inputs explode). The tool refuses to print a truncated false answer past the cap. Each input is limited to 500 digits and 200 numbers per paste so a tab cannot freeze on a malicious wall of text.

Using this least common multiple finder is straightforward:

1. Paste at least two positive integers in the textarea. Separate them with commas, spaces, or new lines. The default is 12, 18, 30.
2. Read the LCM in the result panel. The GCD of the set is listed underneath so you can sanity-check lcm × gcd against a two-number product when you only entered a pair.
3. Copy the LCM, or clear the box. The last list is saved locally for up to 30 days.

Note: zero is rejected. Some programming libraries define lcm(a, 0) = 0; school exercises do not.

Why divide first? a × b can be larger than a language’s naive integer type; a / gcd is always an integer, so (a / gcd) × b is the LCM and stays smaller until the last multiply. This page still uses BigInt, but the order matters for the 8000-digit cap: we check the decimal length of the result, not an overflowed scratch product.

Example - lcm(12, 18)

• gcd(12, 18) = 6 by Euclid: 18 = 1×12 + 6, 12 = 2×6 + 0
• (12 / 6) × 18 = 2 × 18 = 36
• Multiples of 12: 12, 24, 36, 48, …
• Multiples of 18: 18, 36, 54, …

Example - fold in 30

• lcm(12, 18, 30) = lcm(36, 30)
• gcd(36, 30) = 6
• (36 / 6) × 30 = 6 × 30 = 180

Association does not care about order: lcm(12, lcm(18, 30)) is also 180. The calculator reduces left to right.

Two classroom methods sit beside the gcd identity. The prime-power method writes each input in primes and keeps the highest exponent: 12 = 2² × 3, 18 = 2 × 3², 30 = 2 × 3 × 5, so LCM = 2² × 3² × 5 = 180. That is the same number this page computes, and you can inspect the primes on the prime factorization calculator or the individual divisor list on the factor calculator.

The listing-multiples method writes multiples of the largest input until one of them is divisible by every other input. For 12 and 18 you stop at 36. For 12, 18, and 30 you walk 30, 60, 90, 120, 150, 180. Listing is honest and slow. It is a terrible algorithm for 8-digit inputs. This calculator never lists multiples; it only uses Euclid plus the identity.

GCF asks “what is the largest integer that divides all of these?” LCM asks “what is the smallest integer that all of these divide?” For two numbers they lock together: gcd(a, b) × lcm(a, b) = a × b. For 12 and 18, 6 × 36 = 216 = 12 × 18. The identity does not say gcd(a, b, c) × lcm(a, b, c) = a × b × c. Check the default: gcd(12, 18, 30) = 6 and lcm = 180, and 6 × 180 = 1080, while 12 × 18 × 30 = 6480. Do not use the two-argument product formula on a triple.

Word problems tell you which end you want. “When do the blinking lights next coincide?” is LCM. “What is the largest tile that covers both rooms?” is GCF. Adding fractions with unlike denominators uses LCM of the denominators as the common denominator.

When two positive integers share no common prime, they are coprime (gcd = 1) and the identity collapses to lcm(a, b) = a × b. lcm(8, 9) = 72, lcm(7, 15) = 105, lcm(11, 13) = 143. Students who “take the bigger number” because 9 does not look like a multiple of 8 miss those products. Conversely, when one number already divides the other, the LCM is the larger one: lcm(12, 36) = 36, lcm(10, 100) = 100. The default triple is neither of those extremes: 12, 18, and 30 overlap on 2 and 3 but 30 brings a 5, so 180 is bigger than any input and smaller than the product 6480.

lcm(4, 6, 8). lcm(4, 6) = 12 because gcd(4, 6) = 2 and (4/2)×6 = 12. Then lcm(12, 8) = 24 because gcd(12, 8) = 4 and (12/4)×8 = 24. Prime-power check: 4 = 2², 6 = 2×3, 8 = 2³, so LCM = 2³ × 3 = 24. Listing multiples of 8: 8, 16, 24 - and 24 is divisible by 4 and 6.

lcm(21, 6, 35). gcd(21, 6) = 3, lcm = 42. gcd(42, 35) = 7, lcm = 210. Primes: 21 = 3×7, 6 = 2×3, 35 = 5×7 → 2×3×5×7 = 210. This is a standard “when do three warning lights coincide” number.

Adding three fractions. 1/12 + 1/18 + 1/30 = 15/180 + 10/180 + 6/180 = 31/180, which does not simplify further because gcd(31, 180) = 1. If you had used a common denominator of 360 you would still get an equivalent fraction and then have to cancel. The LCM denominator is the one that usually arrives already reduced or close to it.

Traps: (1) Computing gcd and reporting it as the LCM - 6 instead of 180 for the default. (2) Multiplying all three numbers: 6480 is a common multiple, not the least. (3) Taking pairwise LCMs and then multiplying those: lcm(12,18)=36 and lcm(18,30)=90, and 36×90 is 3240, which is not 180. You must fold, not multiply the pairwise answers. (4) Ignoring a repeated number: lcm(12, 12, 18) is still 36; duplicates do not change LCM. (5) Pasting a decimal or a fraction; this page wants whole numbers. LCM of rationals is a different (and rarer) exercise: clear denominators first, take integer LCM, then adjust - not implemented here.

Another trap is overflow-as-silence. A naive program that multiplies a×b before dividing by gcd can wrap a 64-bit integer and print a garbage LCM that still looks like a plausible whole number. BigInt plus the 8000-digit cap is how this page refuses to lie: either you see the exact LCM or you see an error, never a wrapped residue.

GCF can only shrink or stay as you add numbers to the set: gcd(a, b, c) divides gcd(a, b). LCM can only grow or stay: lcm(a, b) divides lcm(a, b, c). Adding a coprime 5 to 12 and 18 multiplies the LCM by 5 (36 → 180) while the GCD stays 6. That is why an LCM calculator needs a digit cap and a GCF calculator usually does not: one output explodes, the other is bounded by the smallest input.

If your homework wants both numbers, compute LCM here and GCF on the sibling page rather than trusting a single product identity on a triple. For two numbers only, you may compute one and recover the other from a × b, which is a useful exam-day check: after you get lcm(12, 18) = 36, 12 × 18 / 36 should return 6.

Folding does not stop at three inputs. lcm(2, 3, 4, 5, 6, 7) is a popular puzzle: lcm(2,3)=6, lcm(6,4)=12, lcm(12,5)=60, lcm(60,6)=60, lcm(60,7)=420. The 6 never increases the running LCM because 6 already divides 60. The 7, being coprime to 60, multiplies it. 420 is also 2^2 × 3 × 5 × 7, the max-exponent product. Paste those six numbers in the textarea to watch the same 420 appear without listing a hundred multiples of 7.

Clock arithmetic is the same fold with named units. Seconds in a minute (60), minutes in an hour (60), and 24 hours are already multiples of smaller integers; LCM questions at that scale are usually “two events every a and b minutes.” If a bell rings every 12 minutes and a chime every 18, they coincide every 36 minutes, twelve times in 432 minutes if they started together - but the first coincidence is still the LCM, 36. A third bell every 30 minutes pushes the first common ring to 180 minutes, the default on this page. That is the pedagogical reason 12, 18, 30 is the example rather than a random triple.

Programming loops that sleep for coprime periods (7 ms and 11 ms) theoretically align every 77 ms. Real-time systems rarely want that; they want a single scheduler tick that is an LCM of the task periods. This calculator will not design your RTOS, but it will tell you that lcm(10, 25, 40) = 200, so a 200-unit tick captures all three periods whole. gcd of that set is 5, shown underneath as the optional check: 10, 25, and 40 are all multiples of 5, and 200 is a multiple of each.

Fractions of an inch in shop drawings (1/8, 1/16, 1/32) use LCM of denominators after you write them as integers 8, 16, 32 - whose LCM is 32, already the finest ruler mark. LCM of 8, 12, 18 is 72, which is why some inch/metric conversion tables look crowded. Convert mixed numbers to improper integers before you paste; this textarea will not parse 1/8.