## Calculator Use

The Least Common Multiple (LCM) is also referred to as the Lowest Common Multiple (LCM) and Least Common Divisor (LCD). For two integers a and b, denoted LCM(a,b), the LCM is the smallest positive integer that is evenly divisible by both a and b. For example, LCM(2,3) = 6 and LCM(6,10) = 30.

The LCM of two or more numbers is the smallest number that is evenly divisible by all numbers in the set.

## Least Common Multiple Calculator

Find the LCM of a set of numbers with this calculator which also shows the steps and how to do the work.

Input the numbers you want to find the LCM for. You can use commas or spaces to separate your numbers. But do not use commas within your numbers. For example, enter **2500, 1000** and not **2,500, 1,000**.

## How to Find the Least Common Multiple LCM

This LCM calculator with steps finds the LCM and shows the work using 6 different methods:

- Listing Multiples
- Prime Factorization
- Cake/Ladder Method
- Division Method
- Using the Greatest Common Factor GCF
- Venn Diagram

## How to Find LCM by Listing Multiples

- List the multiples of each number until at least one of the multiples appears on all lists
- Find the smallest number that is on all of the lists
- This number is the LCM

Example: LCM(6,7,21)

- Multiples of 6: 6, 12, 18, 24, 30, 36,
**42**, 48, 54, 60 - Multiples of 7: 7, 14, 21, 28, 35,
**42**, 56, 63 - Multiples of 21: 21,
**42**, 63 - Find the smallest number that is on all of the lists. We have it in bold above.
- So LCM(6, 7, 21) is 42

## How to find LCM by Prime Factorization

- Find all the prime factors of each given number.
- List all the prime numbers found, as many times as they occur most often for any one given number.
- Multiply the list of prime factors together to find the LCM.

The LCM(a,b) is calculated by finding the prime factorization of both a and b. Use the same process for the LCM of more than 2 numbers.

**For example, for LCM(12,30) we find:**

- Prime factorization of 12 = 2 × 2 × 3
- Prime factorization of 30 = 2 × 3 × 5
- Using all prime numbers found as often as each occurs most often we take 2 × 2 × 3 × 5 = 60
- Therefore LCM(12,30) = 60.

**For example, for LCM(24,300) we find:**

- Prime factorization of 24 = 2 × 2 × 2 × 3
- Prime factorization of 300 = 2 × 2 × 3 × 5 × 5
- Using all prime numbers found as often as each occurs most often we take 2 × 2 × 2 × 3 × 5 × 5 = 600
- Therefore LCM(24,300) = 600.

## How to find LCM by Prime Factorization using Exponents

- Find all the prime factors of each given number and write them in exponent form.
- List all the prime numbers found, using the highest exponent found for each.
- Multiply the list of prime factors with exponents together to find the LCM.

Example: LCM(12,18,30)

- Prime factors of 12 = 2 × 2 × 3 = 2
^{2}× 3^{1} - Prime factors of 18 = 2 × 3 × 3 = 2
^{1}× 3^{2} - Prime factors of 30 = 2 × 3 × 5 = 2
^{1}× 3^{1}× 5^{1} - List all the prime numbers found, as many times as they occur most often for any one given number and multiply them together to find the LCM
- 2 × 2 × 3 × 3 × 5 = 180

- Using exponents instead, multiply together each of the prime numbers with the highest power
- 2
^{2}× 3^{2}× 5^{1}= 180

- 2
- So LCM(12,18,30) = 180

Example: LCM(24,300)

- Prime factors of 24 = 2 × 2 × 2 × 3 = 2
^{3}× 3^{1} - Prime factors of 300 = 2 × 2 × 3 × 5 × 5 = 2
^{2}× 3^{1}× 5^{2} - List all the prime numbers found, as many times as they occur most often for any one given number and multiply them together to find the LCM
- 2 × 2 × 2 × 3 × 5 × 5 = 600

- Using exponents instead, multiply together each of the prime numbers with the highest power
- 2
^{3}× 3^{1}× 5^{2}= 600

- 2
- So LCM(24,300) = 600

## How to Find LCM Using the Cake Method (Ladder Method)

The cake method uses division to find the LCM of a set of numbers. People use the cake or ladder method as the fastest and easiest way to find the LCM because it is simple division.

The cake method is the same as the ladder method, the box method, the factor box method and the grid method of shortcuts to find the LCM. The boxes and grids might look a little different, but they all use division by primes to find LCM.

Find the LCM(10, 12, 15, 75)

- Write down your numbers in a cake layer (row)

Cake/Ladder

10

12

15

75

- Divide the layer numbers by a prime number that is evenly divisible into two or more numbers in the layer and bring down the result into the next layer.

Cake/Ladder

2

10

12

15

75

5

6

- If any number in the layer is not evenly divisible just bring down that number.

Cake/Ladder

5

6

15

75

- Continue dividing cake layers by prime numbers.
- When there are no more primes that evenly divided into two or more numbers you are done.

Cake/Ladder

2

10

12

15

75

3

5

6

15

75

5

5

2

5

25

1

2

1

5

- The LCM is the product of the numbers in the L shape, left column and bottom row. 1 is ignored.
- LCM = 2 × 3 × 5 × 2 × 5
- LCM = 300
- Therefore, LCM(10, 12, 15, 75) = 300

## How to Find the LCM Using the Division Method

Find the LCM(10, 18, 25)

- Write down your numbers in a top table row

DivisionTable

10

18

25

- Starting with the lowest prime numbers, divide the row of numbers by a prime number that is evenly divisible into at least one of your numbers and bring down the result into the next table row.

DivisionTable

2

10

18

25

5

9

- If any number in the row is not evenly divisible just bring down that number.

DivisionTable

2

10

18

25

5

9

25

- Continue dividing rows by prime numbers that divide evenly into at least one number.
- When the last row of results is all 1's you are done.

DivisionTable

2

10

18

25

3

5

9

25

3

5

3

25

5

5

1

25

5

1

1

5

1

1

1

- The LCM is the product of the prime numbers in the first column.
- LCM = 2 × 3 × 3 × 5 × 5
- LCM = 450
- Therefore, LCM(10, 18, 25) = 450

## How to Find LCM by GCF

The formula to find the LCM using the Greatest Common Factor GCF of a set of numbers is:

LCM(a,b) = (a×b)/GCF(a,b)

Example: Find LCM(6,10)

- Find the GCF(6,10) = 2
- Use the LCM by GCF formula to calculate (6×10)/2 = 60/2 = 30
- So LCM(6,10) = 30

A factor is a number that results when you can evenly divide one number by another. In this sense, a factor is also known as a divisor.

The greatest common factor of two or more numbers is the largest number shared by all the factors.

### The greatest common factor GCF is the same as:

- HCF - Highest Common Factor
- GCD - Greatest Common Divisor
- HCD - Highest Common Divisor
- GCM - Greatest Common Measure
- HCM - Highest Common Measure

## How to Find the LCM Using Venn Diagrams

Venn diagrams are drawn as overlapping circles. They are used to show common elements, or intersections, between 2 or more objects. In using Venn diagrams to find the LCM, prime factors of each number, we call the groups, are distributed among overlapping circles to show the intersections of the groups. Once the Venn diagram is completed you can find the LCM by finding the union of the elements shown in the diagram groups and multiplying them together.

## How to Find LCM of Decimal Numbers

- Find the number with the most decimal places
- Count the number of decimal places in that number. Let's call that number D.
- For each of your numbers move the decimal D places to the right. All numbers will become integers.
- Find the LCM of the set of integers
- For your LCM, move the decimal D places to the left. This is the LCM for your original set of decimal numbers.

## Properties of LCM

### The LCM is associative:

LCM(a, b) = LCM(b, a)

### The LCM is commutative:

LCM(a, b, c) = LCM(LCM(a, b), c) = LCM(a, LCM(b, c))

### The LCM is distributive:

LCM(da, db, dc) = dLCM(a, b, c)

### The LCM is related to the greatest common factor (GCF):

LCM(a,b) = a × b / GCF(a,b) and

GCF(a,b) = a × b / LCM(a,b)

## References

[1] Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae, 31st Edition, New York, NY: CRC Press, 2003 p. 101.

[2] Weisstein, Eric W. Least Common Multiple. From *MathWorld*--A Wolfram Web Resource.