What is the least common multiple?

  • In this post, we are going to see what the least common multiple is and how to calculate it.
  • The least common multiple (LCM) of two or more numbers is the smallest number (not counting 0) which is a multiple of all of the numbers.
  • In order to better understand this definition, we are going to look at all of the terms:
  • Multiple: the multiples of a number are the numbers you get when you add a number to itself repeatedly.

What Is the Least Common Multiple?

  • Common Multiple: A common multiple is a number than is a multiple of two or more numbers; in other words, it is a common multiple of all those numbers.

Following with the previous example, we are going to see the common multiples of 2 and 3.

What Is the Least Common Multiple?

You will see that there are common multiples between two and three, which are emphasized in green in the picture: 6 and12.  Keep in mind that the multiples are infinite and we have only shown the first two in this example.

  • Least Common Multiple: the least common multiple is the smallest of the common multiples.

Following with the previous example, if the common multiples of 2 and three 3 were 6 and 12, the least common multiple is 6, since it is smaller than 12.

To continue, we are going to see how to calculate the least common multiple.  You can use two methods.

First method to calculate the least common multiple:

Is what we used before, in other words, we write the first multiples of each number, we note the multiples that are common and we choose the least common multiple.

Second method to calculate the least common multiple:

The first thing we have to do is break down the prime factors of each number.  After, we will have to choose the common factors and not the greatest common to the greatest exponent, and finally, we have to multiple the chosen factors.

We are going to look at an example of this, calculating the LCM of 12 and 8.

What Is the Least Common Multiple?

  1. We are going to break down 12 and 8 into prime factors:
  2. 12 = 22 x 3          8 = 23
  3. Now, we take the highest power of each prime factor in the prime factorization
  4. and multiply them: 23 x 3 = 8 x 3 = 24
  5. So the least common multiple of 12 and 8 is 24.

You can practice some online exercises about LCM and more elementary math in Smartick. Try it for free!

Learn More:

  • Explanation of the Formula to Calculate the Least Common Multiple
  • Review Factoring with Examples
  • Greatest Common Factor (GCF)
  • How to Calculate the Least Common Multiple Using a 100 Square
  • Prime Numbers: Activities with Smartick

Least Common Multiple – Definition with Examples

  • Multiple is a number that can be divided by the given number without leaving a reminder. For example:
  •  20 is a multiple of 5 
  • Or, 5 × 4 = 20
  • And, 20 ÷ 5 = 4

Least common multiple

The least common multiple of two numbers is the “smallest non-zero common number” which is a multiple of both the numbers.

The different methods to find least common multiple of two or more numbers are:

  • Using prime factorization
  • Using repeated division
  • Using multiples

1. LCM using prime factorization

In this method, a factorization tree for each given number is generated by listing the multiples of that number. The last branch of the tree has the least prime factors for that number.

  1. For example, the factorization trees for 36 and 48 are generated as follows:
  2. What Is the Least Common Multiple?
  3. Figure: Prime factorization trees for the number 36 and 48

To find the LCM, pair the common multiples as shown. List them along with the remaining multiples.

What Is the Least Common Multiple?

  • LCM = 2 × 2 × 3 × 3 × 2 × 2
  • LCM = 144
  • 2. LCM using repeated division

In this method, the given numbers are divided by the common divisors until there is no possible further division by the common number. The divisors and the remainders are multiplied together to obtain the LCM.

What Is the Least Common Multiple?

  1. LCM =  2 × 2 × 3 × 4 × 3
  2. LCM = 144
  3. 3. LCM using multiples

To find the LCM using multiples, list the multiples of the numbers in the table as shown. The least common multiple is the first common multiple for the given numbers. 

1 2 3 4 5 6 7 8 9 10 11 12
36 36 72 108 144 180 216 252 288 324 360 396 432
48 48 96 144 192 240 288 336 384 432 480 528 576

For 36 and 48, the number 144 is the LCM.

Application

The dimension of using LCM of two numbers starts with basic math operations such as addition and subtraction on fractional numbers. In math problems where we pair two objects against each other, the LCM value is useful in optimizing the quantities of the given objects. Also, in computer science, the LCM of numbers helps design encoded messages using cryptography.

  Fun facts

  • Greeks used the wax tablets to record multiplication tables in 1st century AD
  • William Oughtred used the symbol “?” for multiplication in the 15th century to teach math

Related math vocabulary

  • Multiples and multiplication
  • Highest Common Factor (HCF)
  • Fractions

What is the lowest common multiple?

We explain what the lowest common multiple is and give examples of how the concept might be taught to your primary school child. Login or Register to add to your saved resources

The lowest common multiple of two numbers is the smallest whole number which is a multiple of both. Teachers may introduce this concept to more able Year 6 children.

Quick reminder: a multiple is a number that can be divided by another number a certain number of times without a remainder.

See also:  Geology: the science of rocks and minerals

What Is the Least Common Multiple?

How to find the lowest common multiple

The basic rule is to list the multiples of the larger number and stop when you get to a multiple of the smaller number. For example:

What is the lowest common multiple of 3 and 8?

Multiples of 3 are 3, 6, 9, 12, 15, 18, 21, 24…

Multiples of 8 are 8, 16, 24, 32, 40…

  • So the lowest common multiple of 3 and 8 is 24.
  • A more difficult challenge may be to ask for the lowest common multiple of a group of three or four numbers.
  • For example:
  • What is the lowest common multiple of 6, 15 and 20?

Least Common Multiple

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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 5 different methods:

  • Listing Multiples
  • Prime Factorization
  • Cake/Ladder Method
  • Division Method
  • Using the Greatest Common Factor GCF

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 = 22 × 31
  • Prime factors of 18 = 2 × 3 × 3 = 21 × 32
  • Prime factors of 30 = 2 × 3 × 5 = 21 × 31 × 51
  • 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
  • Using exponents instead, multiply together each of the prime numbers with the highest power
  • So LCM(12,18,30) = 180

Example: LCM(24,300)

  • Prime factors of 24 = 2 × 2 × 2 × 3 = 23 × 31
  • Prime factors of 300 = 2 × 2 × 3 × 5 × 5 = 22 × 31 × 52
  • 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
  • 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)
  • 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.
  • If any number in the layer is not evenly divisible just bring down that number.
  • Continue dividing cake layers by prime numbers.
  • When there are no more primes that evenly divided into two or more numbers you are done.
  • 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
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How to Find the LCM Using the Division Method

Find the LCM(10, 18, 25)

  • Write down your numbers in a top table row
  • 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.
  • If any number in the row is not evenly divisible just bring down that number.
  • 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.
  • 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 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.

The Math Forum: LCM, GCF.

Least Common Multiple Calculator

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  • Please provide numbers separated by a comma “,” and click the “Calculate” button to find the LCM.

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What is the Least Common Multiple (LCM)?

In mathematics, the least common multiple, also known as the lowest common multiple of two (or more) integers a and b, is the smallest positive integer that is divisible by both. It is commonly denoted as LCM(a, b).

Brute Force Method

There are multiple ways to find a least common multiple. The most basic is simply using a “brute force” method that lists out each integer's multiples.

EX:   Find LCM(18, 26) 18: 18, 36, 54, 72, 90, 108, 126, 144, 162, 180, 198, 216, 23426: 52, 78, 104, 130, 156, 182, 208, 234

As can be seen, this method can be fairly tedious, and is far from ideal.

Prime Factorization Method

A more systematic way to find the LCM of some given integers is to use prime factorization. Prime factorization involves breaking down each of the numbers being compared into its product of prime numbers.

The LCM is then determined by multiplying the highest power of each prime number together. Note that computing the LCM this way, while more efficient than using the “brute force” method, is still limited to smaller numbers.

Refer to the example below for clarification on how to use prime factorization to determine the LCM:

EX:   Find LCM(21, 14, 38) 21 = 3 × 7 14 = 2 × 7 38 = 2 × 19 The LCM is therefore:3 × 7 × 2 × 19 = 798

Greatest Common Divisor Method

A third viable method for finding the LCM of some given integers is using the greatest common divisor. This is also frequently referred to as the greatest common factor (GCF), among other names. Refer to the link for details on how to determine the greatest common divisor.

Given LCM(a, b), the procedure for finding the LCM using GCF is to divide the product of the numbers a and b by their GCF, i.e. (a × b)/GCF(a,b). When trying to determine the LCM of more than two numbers, for example LCM(a, b, c) find the LCM of a and b where the result will be q. Then find the LCM of c and q.

The result will be the LCM of all three numbers. Using the previous example:

EX:   Find LCM(21, 14, 38) GCF(14, 38) = 2

LCM(14, 38) =    = 266

GCF(266, 21) = 7

LCM(266, 21) =    = 798

LCM(21, 14, 38) = 798

Note that it is not important which LCM is calculated first as long as all the numbers are used, and the method is followed accurately. Depending on the particular situation, each method has its own merits, and the user can decide which method to pursue at their own discretion.

Lowest Common Multiples

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Common multiples are multiples that two numbers have in common. These can be useful when working with fractions and ratios.

Example:What are some common multiples of 2 and 3?

Multiples of 2: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24

Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27…Common multiples of 2 and 3 include 6, 12, 18, and 24.Example:What are some common multiples of 25 and 30?

Multiples of 25: 25, 50, 75, 100, 125, 150, 175, 200, 225, 250, 275, 300, 325…

Multiples of 30: 30, 60, 90, 120, 150, 180, 210, 240, 270, 300, 330…Common multiples of 25 and 30 include 150 and 300.

The lowest common multiple or least common multiple is the lowest multiple two numbers have in common.

There are two ways of finding the lowest common multiple of two numbers.

Method 1: Listing Multiples

The first way to find the lowest common multiple is to do what we did above: write out a list of the lowest multiples of each number, and look for the lowest multiple both numbers have in common.

Example:What is the lowest common multiple of 2 and 3?

Multiples of 2: 2, 4, 6, 8…

Multiples of 3: 3, 6, 9…The lowest common multiple of 2 and 3 is 6.Example:What is the lowest common multiple of 25 and 30?

Multiples of 25: 25, 50, 75, 100, 125, 150, 175…

Multiples of 30: 30, 60, 90, 120, 150, 180…The lowest common multiple of 25 and 30 is 150.

Method 2: Factors

The other way to find the lowest common multiple is to list the prime factors for each number. Remove the prime factors both numbers have in common. Multiply one of the numbers by the remaining prime factors of the other number. The result will be the lowest common multiple.

Example:What is the lowest common multiple of 25 and 30?

The prime factors of 25 are 5 x 5.

The prime factors of 30 are 2 x 3 x 5. Remove the 5 that 25 and 30 have in common as a prime factor. Multiply 25 by the remaining prime factors of 30. 25 x 2 x 3 = 150.

The lowest common multiple of 25 and 30 is 150.

You'll get the same results no matter which number you work with:

Example:What is the lowest common multiple of 25 and 30?

The prime factors of 25 are 5 x 5.

The prime factors of 30 are 2 x 3 x 5. Remove the 5 that 25 and 30 have in common as a prime factor. Multiply 30 by the remaining prime factors of 25. 30 x 5 = 150.

The lowest common multiple of 25 and 30 is 150.

Another Example

Example:What is the lowest common multiple of 42 and 48?

The prime factors of 42 are 2 x 3 x 7.

The prime factors of 48 are 2 x 2 x 2 x 2 x 3. Remove the 2 x 3 that 42 and 48 have in common as prime factors. Multiply 48 by the remaining prime factors of 42. 48 x 7 = 336.

The lowest common multiple of 42 and 48 is 336.

What if they have no prime factors in common?

Example:What is the lowest common multiple of 44 and 45? The prime factors of 44 are 2 x 2 x 11. The prime factors of 45 are 3 x 3 x 5. 44 and 45 have no prime factors in common.

  • Either:
  • Or:
  • Or:
  • The lowest common multiple of 44 and 45 is 1980.

Multiply 44 by the remaining prime factors of 45. 44 x 3 x 3 x 5 = 1980. Multiply 45 by the remaining prime factors of 44. 45 x 2 x 2 x 11 = 1980. 44 x 45 = 1980.

As that last example illustrates, if two numbers have no prime factors in common, the lowest common multiple will be equal to the product of the two numbers.

Treat primes as prime factors

If one number is prime, you can treat it as its own prime factor.

Example:What is the lowest common multiple of 7 and 30? 7 is a prime number. The prime factors of 30 are 2 x 3 x 5. 7 and 30 have no prime factors in common. 7 x 30 = 210.

The lowest common multiple of 7 and 30 is 210.

Example:What is the lowest common multiple of 2 and 3? 2 is a prime number. 3 is a prime number. 2 and 3 have no prime factors in common. 2 x 3 = 6.

The lowest common multiple of 2 and 3 is 6.

Example:What is the lowest common multiple of 3 and 30?

3 is a prime number.

The prime factors of 30 are 2 x 3 x 5. Remove the 3 that 3 and 30 have in common as a prime factor.Either: Multiply 3 by the remaining prime factors of 30. 3 x 2 x 5 = 30

Or:

You would normally multiply 30 by the remaining prime factors of 3, but there are no remaining prime factors.The lowest common multiple is 30.

Prime results

As you can see from the above, there are two scenarios if at least one number is prime:

  • If one number is prime, and the other number's prime factors include that prime number, the lowest common multiple will be equal to the non-prime number.
  • If one number is prime, and the other number's prime factors do not include that prime number, the lowest common multiple will be equal to the product of the two numbers.

(The second scenario also includes cases where both numbers are prime.)

Common Factors Mixed Numbers and Improper Fractions .com/ipa/0/9/3/3/3/5/A0933352.html

  • Reducing Fractions to Lowest Terms

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