Least Common Multiple Calculator

What this calculator does

So, you have two or more whole numbers and you want the smallest number that all of them divide into evenly. That is the least common multiple, and this tool finds it. Enter your numbers and it returns their least common multiple.

You type the numbers separated by commas or spaces, so it handles two numbers or a longer list in one go.

How to use it

1. Enter your whole numbers, separated by commas or spaces, like 4, 6 or 4 6.
2. Press Calculate.

What the least common multiple is

A multiple of a number is what you get by multiplying it by 1, 2, 3, and so on. The multiples of 4 are 4, 8, 12, 16, and onward. A common multiple of several numbers is a number that appears in all their multiplication tables. The least common multiple is the smallest such number. For 4 and 6, the common multiples include 12, 24, 36, and the smallest of them is 12, so the least common multiple of 4 and 6 is 12.

How this tool finds it: prime powers

Rather than list out endless multiples, the tool uses a cleaner method built on prime factors. It breaks each number down into its prime factors, noting how many times each prime appears. Then, for every prime that shows up in any of the numbers, it takes the highest power of that prime seen in any single number, and multiplies all those highest powers together. The result is the least common multiple. This works because the least common multiple has to contain enough of each prime to cover every number, and the highest power is exactly enough.

The partner of the greatest common factor

The least common multiple and the greatest common factor are two sides of one coin. The greatest common factor works downward, finding the largest number that divides into your numbers; the least common multiple works upward, finding the smallest number they divide into. For any two numbers, there is a neat link between them: the greatest common factor multiplied by the least common multiple equals the two numbers multiplied together. So if you know one, you can find the other, which is why these two tools are natural companions.

Where the LCM is useful

The least common multiple is the key to adding fractions with different denominators. To add them, you rewrite each over a common denominator, and the smallest one that works is the least common multiple of the denominators, which keeps the numbers as small as possible. It also answers timing puzzles, the kind where two events repeat on different cycles and you want to know when they next line up. If one thing happens every 4 days and another every 6, they coincide every 12 days, their least common multiple.

A worked example

Enter 4 and 6. The tool factors 4 into 2 times 2, and 6 into 2 times 3. The primes involved are 2 and 3. The highest power of 2 in either number is 2 times 2, from the 4, and the highest power of 3 is a single 3, from the 6. Multiplying these highest powers, 2 times 2 times 3, gives 12. So the least common multiple of 4 and 6 is 12.

Questions people ask

What is the least common multiple?

The smallest number that all your numbers divide into evenly. For 4 and 6 it is 12.

How does the tool find it?

It breaks each number into prime factors and multiplies together the highest power of each prime that appears in any number.

How does it relate to the greatest common factor?

They are partners. For two numbers, the greatest common factor times the least common multiple equals the product of the numbers.

Why is it useful for fractions?

It gives the smallest common denominator for adding fractions, keeping the numbers as small as possible.

How does it solve timing problems?

Two repeating cycles next coincide after a number of steps equal to the least common multiple of their periods.

References

On the least common multiple. The least common multiple is built from the highest power of each prime, and it pairs with the greatest common divisor.

1. Eric W. Weisstein, "Fundamental Theorem of Arithmetic," from MathWorld, a Wolfram resource, on the unique prime factorization that the prime-power method relies on.
2. Eric W. Weisstein, "Greatest Common Divisor," from MathWorld, a Wolfram resource, on the companion quantity to the least common multiple.