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The LCM is built from every prime factor across your numbers, each one taken at its highest power.
The least common multiple, almost always shortened to LCM, is the smallest positive whole number that every number in a set divides into with no remainder. If you list the multiples of each number, the results of multiplying it by 1, 2, 3, and so on, the LCM is the first value that shows up in every one of those lists at the same time. Because it must be divisible by each original number, the LCM is always greater than or equal to the largest number in the set, and it is unique for any given set of numbers.
Among all the numbers that every input divides into evenly, the LCM is specifically the smallest one. There are infinitely many common multiples of any set of numbers. You could always multiply the LCM by 2, 3, or any whole number to get another one, but only one of them is the least.
By definition, dividing the LCM by any of the original numbers always produces a whole number with no remainder. This is the property that makes the LCM useful for adding fractions: once every denominator divides evenly into a shared number, each fraction can be rewritten over that number without changing its value.
The LCM is not limited to pairs. You can find the least common multiple of three, four, or more numbers by combining them two at a time, or by factoring all of them into primes at once and taking the highest power of each prime that appears anywhere in the set.
There are three standard ways to find a least common multiple by hand. Listing multiples is the most intuitive and works well for small numbers. Prime factorization scales better as numbers grow larger or as you add more numbers to the set. The GCF formula is the fastest route whenever you already know (or can quickly compute) the greatest common factor of two numbers.
| Method | Best used when | How it works |
|---|---|---|
| Listing multiples | Numbers are small and easy to count through by hand | Write out multiples of each number until one repeats across every list |
| Prime factorization | Numbers are larger or you have three or more numbers | Factor each number into primes, then multiply the highest power of each prime |
| GCF formula | You already know (or can find) the greatest common factor | Multiply the two numbers and divide by their GCF |
This method is the most visual way to understand what LCM actually means. Write down the multiples of each number in order, first, second, third, and so on, until a value appears on every list. That shared value is the LCM. For small numbers this usually only takes a handful of steps, and it builds strong intuition for why the LCM has to be divisible by every original number.
For larger numbers, listing multiples gets tedious. Prime factorization scales much better: break each number down into a product of primes, then build the LCM by taking the highest power of every prime that shows up anywhere in the set. This is the same method the calculator above uses to build its step-by-step breakdown.
Every pair of numbers has a mathematical relationship between its LCM and its GCF (greatest common factor): multiplying the two numbers together and dividing by their GCF always gives the LCM. This is usually the fastest method by hand whenever the GCF is easy to spot.
The least common denominator, or LCD, is not a different calculation from the LCM. It is the exact same calculation applied specifically to the denominators of a set of fractions. Whenever you add or subtract fractions with different denominators, the LCD you need is simply the LCM of those denominators. Once the fractions share that denominator, adding or comparing them is a matter of ordinary whole-number arithmetic on the numerators.
Take 1/4 and 1/6 as an example. Their denominators are 4 and 6. The LCM of 4 and 6 is 12, so the LCD of 1/4 and 1/6 is also 12. Rewritten over that denominator, 1/4 becomes 3/12 and 1/6 becomes 2/12, and now they can be added directly to get 5/12.
A general-purpose calculation for any set of whole numbers, used in scheduling, packaging, gear ratios, and anywhere repeating quantities need to be lined up.
The LCM applied specifically to the denominators of a set of fractions, so those fractions can be rewritten with a shared denominator and combined arithmetically.
Six fully solved examples below cover the listing-multiples method, prime factorization, and LCM problems with three numbers at once, a good cross-section of the situations you're likely to run into.
Look up the least common multiple of any pair of numbers from 2 through 10 in the grid below. Find one number along the top and the other down the side. The cell where the row and column meet is their LCM.
| LCM | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|---|---|
| 2 | 2 | 6 | 4 | 10 | 6 | 14 | 8 | 18 | 10 |
| 3 | 6 | 3 | 12 | 15 | 6 | 21 | 24 | 9 | 30 |
| 4 | 4 | 12 | 4 | 20 | 12 | 28 | 8 | 36 | 20 |
| 5 | 10 | 15 | 20 | 5 | 30 | 35 | 40 | 45 | 10 |
| 6 | 6 | 6 | 12 | 30 | 6 | 42 | 24 | 18 | 30 |
| 7 | 14 | 21 | 28 | 35 | 42 | 7 | 56 | 63 | 70 |
| 8 | 8 | 24 | 8 | 40 | 24 | 56 | 8 | 72 | 40 |
| 9 | 18 | 9 | 36 | 45 | 18 | 63 | 72 | 9 | 90 |
| 10 | 10 | 30 | 20 | 10 | 30 | 70 | 40 | 90 | 10 |