LCM Calculator

Our online LCM calculator helps you quickly find the least common multiple (LCM) of two or more numbers. The least common multiple is the smallest positive integer that is a multiple of all the given numbers. For instance, the LCM of 2 and 3 is 6, because 6 is the smallest number evenly divisible by both 2 and 3. Beyond providing instant results, this lowest common multiple calculator offers step-by-step solutions using various popular methods: listing multiples, prime factorization, the cake/ladder method, the division method, the GCF method, and Venn diagrams.

Directions for Use

• To use the LCM calculator, simply enter your numbers and click “Calculate.”
• Separate your numbers using spaces or commas. Please note that commas cannot be used within a single number. For example, you should write one thousand as 1000, not 1,000. The tool will instantly calculate the least common multiple of your inputted numbers.
• To view a step-by-step solution, select your preferred calculation method from the drop-down menu and click “Calculate.”
• If you wish to see the solution steps using a different method, just select an alternative option from the drop-down menu and click “Calculate” again.

Calculation Algorithms

Listing multiples

The most straightforward way to find the least common multiple of several numbers is to write out a list of multiples for each number until you find a common multiple that appears on every list. This matching number is your LCM.

For example, let’s find the LCM of 5 and 7, denoted as LCM (5, 7):

Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, etc.

Multiples of 7: 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, etc.

Since 35 is the first multiple that appears in both lists, LCM (5, 7) = 35.

Prime factorization

To find the LCM of multiple numbers using prime factorization, follow these simple steps:

1. Write down the prime factors for each number.
2. Express the prime factorization of each number in exponent form (for instance, 2 × 2 × 2 becomes 2³).
3. Multiply the highest powers of all the prime factors present.
4. The resulting product is the LCM of the given numbers.

Note that you can also find the LCM without expressing the prime factors in exponent form. In this case, simply replace step 3 by multiplying each prime factor the maximum number of times it occurs in the factorization of any one given number.

For example, let’s find the LCM of 3, 12, and 40, written as LCM (3, 12, 40):

1. Finding prime factors of each number.

Prime factors of 3: 3 is prime.

Prime factors of 12: 2 × 2 × 3

Prime factors of 40: 2 × 2 × 2 × 5

2. Writing prime factorization in exponent form.

3 = 3¹

12 = 2² × 3

40 = 2³ × 5¹

3. Multiplying the highest powers of all prime factors.

2³ × 3¹ × 5¹ = 120

4. LCM (3, 12, 40) = 120

Without using exponent form, step 3 would look like this: 2 × 2 × 2 × 3 × 5 = 120.

Our LCM calculator demonstrates both of these prime factorization variations in its step-by-step solutions.

Cake/Ladder

This method gets its name because the resulting calculation resembles a layered cake (or a ladder). Let’s explore this algorithm by walking through an example to find the LCM of 12, 15, and 24.

1. First, write your numbers horizontally and draw a “ladder step” or a “cake layer” around them, like this:

2. Find a prime number that evenly divides at least two of the given numbers. Write this divisor to the left, perform the division, and drop the results down into the next “cake layer.” If a number is not evenly divisible, simply bring it down as is.

Let’s use 2 as our first divisor, since both 12 and 24 are divisible by 2. This gives us the following setup:

3. Continue repeating step 2 until there are no numbers left that can evenly divide at least two of the bottom numbers:

4. The LCM is the product of the numbers in the left column and the numbers remaining in the bottom row. For our example:

LCM (12, 15, 24) = 2 × 2 × 3 × 1 × 5 × 2 = 120

Division method

The division method is quite similar to the cake/ladder method. However, in this approach, you continue performing divisions as long as any single one of the given numbers is divisible by a prime number. Ultimately, the bottom row will consist entirely of ones, and you calculate the LCM by multiplying all the prime divisors listed in the left column. Using our previous example for finding the LCM (12, 15, 24), the division table looks like this:

And finally, LCM (12, 15, 24) = 2 × 2 × 2 × 3 × 5 = 120

GCF method

To find the lowest common multiple of two numbers using their Greatest Common Factor (GCF), you can apply the following formula:

LCM (x, y) = (x × y) / GCF (x, y)

To find the LCM of more than two numbers, you simply iterate this formula. For instance, the LCM of three numbers is calculated like this:

LCM (x, y, z) = LCM (LCM (x, y), z)

As an example, let’s find the LCM of 6 and 8. The GCF (6, 8) is 2. Therefore:

LCM (6, 8) = (6 × 8) / 2 = 48 / 2 = 24

Venn diagram

To calculate the LCM using a Venn diagram, start by identifying the prime factors for each number. Next, group these factors based on their intersection with two or three of the given numbers, and map them out in the diagram. For LCM (12, 15, 24), the Venn diagram looks like this:

Please note: Our online LCM calculator generates visual Venn diagram solutions exclusively for sets of 2 or 3 numbers.

Calculation example

Mike and Lina both attend karate lessons, but on different schedules: Mike goes every 5 days, while Lina goes every 3 days. If they attended a lesson together today, how many days will it take for them to have a class together again?

Solution

To solve this practical problem, we need to find the least common multiple of 5 and 3, written as LCM (5, 3). Let’s calculate this using the prime factorization method.

3 is a prime number, therefore 3 = 3¹

5 is also a prime number, therefore 5 = 5¹

LCM (5, 3) = 3¹ × 5¹ = 15

Answer

Mike and Lina will attend a karate lesson together again in exactly 15 days.