What is the LCM of and 9?

The first step to this method of finding the Least Common Multiple of 7 and 9 is to begin to list a few multiples for each number. If you need a refresher on how to find the multiples of these numbers, you can see the walkthroughs in the links below for each number.

Let’s take a look at the multiples for each of these numbers, 7 and 9:

What are the Multiples of 7?

What are the Multiples of 9?

Let’s take a look at the first 10 multiples for each of these numbers, 7 and 9:

First 10 Multiples of 7: 7, 14, 21, 28, 35, 42, 49, 56, 63, 70

First 10 Multiples of 9: 9, 18, 27, 36, 45, 54, 63, 72, 81, 90

You can continue to list out the multiples of these numbers as long as needed to find a match. Once you do find a match, or several matches, the smallest of these matches would be the Least Common Multiple. For instance, the first matching multiple(s) of 7 and 9 are 63, 126, 189. Because 63 is the smallest, it is the least common multiple.

The LCM of 7 and 9 is 63.

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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.

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:

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:

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.

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 = 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

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 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.