How to find the Least Common Multiple Worksheet, Examples, and Definition

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4 EASY steps to learn how to find LCM (Avoid these common mistakes)

Key Points about Least Common Multiple

  • LCM is a mathematical concept used to find the smallest common multiple of two or more numbers.
  • To find the LCM, one can list the multiples of each number and identify the smallest multiple that is common to all of them.
  • Understanding LCM is important for developing problem-solving skills and mastering more advanced mathematical concepts

What is the Least Common Multiple?

The Least Common Multiple (LCM) is a mathematical concept used to find the smallest common multiple of two or more numbers. In math, numbers are often used to represent real-world quantities, and finding the LCM helps to simplify mathematical operations involving those numbers.

To find the LCM of two or more numbers, one can list the multiples of each number and then identify the smallest multiple that is common to all of them. This process is often used in elementary math classes and is an important skill for solving more complex problems in algebra and other branches of mathematics.

Understanding the concept of LCM is important for students of all ages, as it lays the foundation for more advanced mathematical concepts. By mastering the techniques for finding the LCM of two or more numbers, students can develop a deeper understanding of mathematical operations and improve their problem-solving skills.

The Least Common Multiple is the smallest number is that a multiple of both original numbers. A common multiple is any number that is a multiple of both original numbers. One way to find the Least Common Multiple is to list out the multiples of each number until you find a number that is a multiple of both numbers. Another way to find the Least Common Multiple is to factor each number into prime factors. Then you multiply the largest set of each factor together with all the other largest sets of each factor.

Common Core Standard: 6.NS.4
Related Topics: Adding Decimals, Subtracting Decimals, Multiplying Decimals, Dividing Decimals, Greatest Common Factor
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How to find the Least Common Multiple

Least Common Multiple Definition

The least common multiple (LCM) is the smallest positive integer that is a multiple of two or more numbers. In other words, it is the lowest common multiple of a set of numbers. The LCM is also known as the lowest common denominator or smallest common multiple.

To find the LCM of two or more numbers, you need to identify their common multiples, which are the multiples that they share. The LCM is then the smallest of these common multiples. For example, the LCM of 3 and 4 is 12, because 12 is the smallest number that is a multiple of both 3 and 4.

The LCM is an important concept in mathematics, particularly in number theory and algebra. It is used in various applications, such as simplifying fractions, adding and subtracting fractions with different denominators, and solving equations involving fractions.

Calculating the LCM can be done using different methods, such as the prime factorization method, the listing method, or the division method. The prime factorization method involves finding the prime factors of each number and multiplying them together, while the listing method involves listing the multiples of each number until a common multiple is found. The division method involves dividing each number by their greatest common divisor (GCD) and multiplying the quotients together.

In conclusion, the least common multiple is the smallest positive integer that is a multiple of two or more numbers. It is an essential concept in mathematics and has various applications in different fields.

 

Finding Least Common Multiple

To find the least common multiple (LCM) of a set of numbers, there are different methods that can be used. Two popular methods are the prime factorization method and the division method.

Finding Least Common Multiple Using Prime Factorization in 3 Easy Steps

The prime factorization method involves finding the prime factors of each number in the set and multiplying the highest power of each prime factor together. For example, to find the LCM of 12 and 18:

  1. Find the prime factors of each number:
    • 12 = 2 x 2 x 3
    • 18 = 2 x 3 x 3
  2. Write the prime factors with the highest power:
    • 12 = 2^2 x 3
    • 18 = 2 x 3^2
  3. Multiply the prime factors together:
    • LCM = 2^2 x 3^2 = 36

Therefore, the LCM of 12 and 18 is 36.

Finding LCM Using Division Method in 4 Quick Steps

The division method involves dividing each number in the set by the smallest prime number that divides at least one of the numbers. The quotients are then divided by the smallest prime number that divides at least one of the remaining numbers, and so on, until the only remaining numbers are all prime. The LCM is the product of all the divisors and the remaining prime numbers.

For example, to find the LCM of 12 and 18:

  1. Divide by the smallest prime number that divides at least one of the numbers:
    • 12 ÷ 2 = 6
    • 18 ÷ 2 = 9
  2. Divide by the smallest prime number that divides at least one of the remaining numbers:
    • 6 ÷ 3 = 2
    • 9 ÷ 3 = 3
  3. The only remaining numbers are prime: 2 and 3.
  4. Multiply all the divisors and the remaining prime numbers:
    • LCM = 2 x 2 x 3 x 3 = 36

Therefore, the LCM of 12 and 18 is 36.

Both methods can be used to find the LCM of any set of numbers. The prime factorization method is usually faster for smaller numbers, while the division method is more efficient for larger numbers. Calculating the LCM is important in many areas of mathematics, such as algebra and number theory.

 

 

How to find the Least Common Multiple Solution

Solving LCM Examples in 3 Easy Steps

  1. Find the Prime Factorization of each number.
  2. List the Prime Factors of each number out so you can see each type of factor.
  3. Multiply the largest set of each factor together with the other factors.

HCF and LCM Examples

The least common multiple (LCM) is the smallest multiple that two or more numbers have in common. For example, the LCM of 4 and 6 is 12, because 12 is the smallest multiple that both 4 and 6 have in common. The highest common factor (HCF) is the largest factor that two or more numbers have in common. For example, the HCF of 4 and 6 is 2, because 2 is the largest factor that both 4 and 6 have in common.

To find the LCM and HCF of two or more numbers, you can use the prime factorization method. First, find the prime factors of each number. Then, multiply the common prime factors to find the LCM, and multiply the highest power of each common prime factor to find the HCF.

For example, to find the LCM and HCF of 12 and 18:

  • The prime factors of 12 are 2 x 2 x 3
  • The prime factors of 18 are 2 x 3 x 3

The common prime factors are 2 and 3. To find the LCM, multiply the common prime factors and the unique prime factors: 2 x 2 x 3 x 3 = 36. To find the HCF, multiply the highest power of each common prime factor: 2 x 3 = 6. Therefore, the LCM of 12 and 18 is 36, and the HCF is 6.

GCF and LCM Examples

The greatest common factor (GCF) is the largest factor that two or more numbers have in common. The LCM is the smallest multiple that two or more numbers have in common. To find the LCM and GCF of two or more numbers, you can use the prime factorization method.

For example, to find the LCM and GCF of 24 and 36:

  • The prime factors of 24 are 2 x 2 x 2 x 3
  • The prime factors of 36 are 2 x 2 x 3 x 3

The common prime factors are 2, 2, and 3. To find the LCM, multiply the common prime factors and the unique prime factors: 2 x 2 x 2 x 3 x 3 = 72. To find the GCF, multiply the highest power of each common prime factor: 2 x 2 x 3 = 12. Therefore, the LCM of 24 and 36 is 72, and the GCF is 12.

In summary, the LCM is the smallest multiple that two or more numbers have in common, while the HCF is the largest factor that two or more numbers have in common. To find the LCM and HCF of two or more numbers, you can use the prime factorization method. Similarly, the GCF is the largest factor that two or more numbers have in common, and you can also use the prime factorization method to find the LCM and GCF of two or more numbers.

 

5 Quick Least Common Multiple Practice Problems

/5

Least Common Multiple Quiz

Click Start to begin the practice quiz!

1 / 5

Find the Least Common Multiple for the pair of numbers below.

25 & 80

2 / 5

Find the Least Common Multiple for the pair of numbers below.

56 & 20

3 / 5

Find the Least Common Multiple for the pair of numbers below.

50 & 20

4 / 5

Find the Least Common Multiple for the pair of numbers below.

12 & 40

5 / 5

Find the Least Common Multiple for the pair of numbers below.

48 & 80

Your score is

0%

 

What is LCM in Math?

In mathematics, LCM stands for Least Common Multiple. It is the smallest positive integer that is divisible by two or more numbers without a remainder. For example, the LCM of 4 and 6 is 12, because 12 is the smallest number that is divisible by both 4 and 6.

LCM is an important concept in math, particularly in number theory and algebra. It is used in various mathematical operations such as addition, subtraction, and comparison of fractions. It is also used to solve problems related to ratios, proportions, and percentages.

To find the LCM of two or more numbers, there are different methods that can be used, including the prime factorization method, the listing multiples method, and the Euclidean algorithm. Each method has its advantages and disadvantages, and the choice of method depends on the specific problem and the numbers involved.

Overall, LCM is a fundamental concept in mathematics that has many applications in various fields, including science, engineering, and economics. It is an important tool for solving problems that involve multiples, factors, and fractions, and it is essential for understanding more advanced topics in math.

 

How to Find LCM of Two Numbers

To find the Least Common Multiple (LCM) of two numbers, there are various methods that can be used. Here are a few popular methods:

Method 1: Listing Multiples

One of the simplest methods to find the LCM of two numbers is by listing their multiples and finding the smallest multiple that is common to both numbers. For example, to find the LCM of 4 and 6:

  • Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100
  • Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 96, 102, 108, 114, 120

The smallest multiple that is common to both 4 and 6 is 12. Therefore, the LCM of 4 and 6 is 12.

Method 2: Prime Factorization

Another method to find the LCM of two numbers is by using prime factorization. To use this method, first, find the prime factors of each number. Then, write the prime factors of each number in exponential form. Finally, multiply the highest power of each prime factor together.

For example, to find the LCM of 12 and 18:

  • Prime factors of 12: 2 x 2 x 3
  • Prime factors of 18: 2 x 3 x 3

The highest power of 2 is 2², the highest power of 3 is 3². Therefore, the LCM of 12 and 18 is 2² x 3² = 36.

Method 3: Using the GCD

The LCM of two numbers can also be found by using the Greatest Common Divisor (GCD) of the two numbers. To use this method, first, find the GCD of the two numbers. Then, divide the product of the two numbers by their GCD.

For example, to find the LCM of 15 and 25:

  • GCD of 15 and 25: 5
  • Product of 15 and 25: 375

375 ÷ 5 = 75. Therefore, the LCM of 15 and 25 is 75.

By using any of these methods, you can find the LCM of any two numbers.

 

How to Find LCM of 3 Numbers

Finding the LCM of three numbers is similar to finding the LCM of two numbers. However, it requires an additional step. Here are the steps to find the LCM of three numbers:

  1. Find the LCM of the first two numbers using the LCM formula or the prime factorization method.
  2. Find the LCM of the result from step 1 and the third number using the same method.

For example, let’s find the LCM of 4, 6, and 8 using both methods.

LCM Formula Method

The LCM formula is LCM(a, b) = (a * b) / GCD(a, b), where GCD is the greatest common divisor. Here are the steps:

  1. Find the GCD of the first two numbers: GCD(4, 6) = 2.
  2. Use the formula to find the LCM of 4 and 6: LCM(4, 6) = (4 * 6) / 2 = 12.
  3. Find the GCD of 12 and 8: GCD(12, 8) = 4.
  4. Use the formula to find the LCM of 4, 6, and 8: LCM(4, 6, 8) = (12 * 8) / 4 = 24.

Therefore, the LCM of 4, 6, and 8 is 24.

Prime Factorization Method

Here are the steps to find the LCM of 4, 6, and 8 using the prime factorization method:

  1. Find the prime factorization of each number: 4 = 2 * 2, 6 = 2 * 3, 8 = 2 * 2 * 2.
  2. Identify the highest power of each prime factor: 2^2, 3^1, 2^3.
  3. Multiply the highest powers: 2^3 * 3^1 = 24.

Therefore, the LCM of 4, 6, and 8 is 24.

In conclusion, finding the LCM of three numbers requires an additional step but follows the same principles as finding the LCM of two numbers. The LCM formula and prime factorization method are both effective ways to find the LCM of three numbers.

 

LCM FAQ

What is the difference between LCM and GCD?

The LCM (Least Common Multiple) is the smallest number that is a multiple of two or more numbers, while the GCD (Greatest Common Divisor) is the largest number that divides two or more numbers without leaving a remainder.

How do you find the LCM of two numbers?

To find the LCM of two numbers, you can list the multiples of each number until you find the smallest multiple that they have in common. Alternatively, you can use the prime factorization method, where you find the prime factors of each number and then multiply the highest power of each prime factor together.

What is the LCM of prime numbers?

The LCM of two prime numbers is the product of those two numbers. For example, the LCM of 3 and 5 is 15.

Can LCM of three numbers be zero?

No, the LCM of three numbers cannot be zero. The LCM is always a positive number, and it is only zero if one or more of the numbers is zero.

What is the LCM of fractions?

To find the LCM of fractions, you need to find the LCM of the denominators and then use that as the denominator for the resulting fraction. For example, the LCM of 2/3 and 3/4 is 12, so you can convert the fractions to 8/12 and 9/12, respectively.

What is the LCM of decimals?

Decimals are not typically used in LCM calculations, as LCM is usually used for whole numbers. However, if you need to find the LCM of decimals, you can convert them to fractions and then find the LCM of the resulting fractions.

How to do Least Common Multiple?

To find the LCM of two or more numbers, you can use either the listing method or the prime factorization method. The listing method involves listing the multiples of each number until you find the smallest multiple that they have in common. The prime factorization method involves finding the prime factors of each number and then multiplying the highest power of each prime factor together.

 

Least Common Multiple Worksheet Video Explanation

Watch our free video on finding Least Common Multiple. This video shows how to solve problems that are on our free Least Common Multiples worksheet that you can get by submitting your email above.

Watch the free Least Common Multiple video on YouTube here: Least Common Multiple Video

Video Transcript:

This video is about how to find least common multiple​. You can get the lowest common multiple worksheet used in this video for free by clicking on the link in the description below. The least common multiple definition states that the least common multiple is the smallest multiple that both numbers have in common. We’re going to do a quick example finding the least common multiple meaning between the numbers 4 and 6. Now I can take both 4 and 6 and I can list out the multiples of each number. We’re going to start with 4. 4 is a multiple 8 is a multiple 12. 4 times 4 16 20 and then 24. We’ll go to and then I can do the same thing for 6. We start with 6 and then we go to 12, then we go to 18 but I don’t have to go any further because I can look and I can see that the number 12 is a multiple of four and six and it is the smallest multiple that they have in common. I know that the least common multiple between four and six would be 12.

You can do the same thing for larger numbers by using factor trees and that’s what I’m going to do for finding the least common multiple between 50 and 20. I’m going to first take 50 and I’m going to break it down by prime factorization. Prime factorization is sometimes shown as a factor tree which is what I’m going to do here. What we’re going to do is we’re going to take each number and break it down into prime factors. The number 50 I know is even so I’m going to divide by 2 2 times 25 gives us 50. Now I know 2 is a prime number. I don’t have to break it down any further but 25 can be broken down into five times five. The fives are also prime numbers which means they cannot be broken down any further. 50 is done because all that we have are prime numbers we’re going to do the same thing for number 20. 20 is an even number I’m going to start with 2. 2 times 10 is 20. 10 is also an even number. i’m going to use 2 again and 2 times 5 is 10. 2 and 5 are also prime numbers so I know that 20 has been broken down completely into prime numbers. I’m going to highlight all of my factors for 50 and all of my factors for 20. Now what I’m going to do is I’m going to take both numbers and I’m going to list them out in their prime factorization. 50 is 2 times 5 times 5 and 20 is 2 times 2 times five. Now when you are looking for your least common multiple, you’re going to use each factor and you’re going to use the largest amount of each factor. When you’re finding the least common multiple you have to use each prime number and you have to use the largest amount of each prime number. For example, if you look, we have both the number 2 and the number 5 is our prime numbers that we have from our factoring. We have to use two when we multiply them all together to get the least common multiple and we also have to use five now we’re going to use two twos because 20 has two twos in it, which is the largest amount of twos, and then we’re going to use five times five because 50 has two fives in it now 50 also has a single 2 and 20 has a single five but we don’t use these because we’ve already used our prime factor of two and our prime factor of five and we took two twos from 20 and two fives from 50. Now to get our answer we just multiply two times two which is four. Four times five is twenty, twenty times five is one hundred. Our least common multiple between fifty and twenty is going to be one hundred.

Let’s do a couple practice problems from our common multiples worksheet. Number one gives us the numbers 12 and 40 and asked us to find the least common multiple between the two. I’m going to start by factoring the number 12 and the number 40. When I factor 12, 12 is an even number. I’m going to start with 2, 2 times 6 is 12, then 6 is also an even number. I’m going to use 2 again and then 2 times 3 is 6. Now 12 has been factored into prime numbers. 2 2 and 3 are all prime I’m going to do the same thing for 40. I’m going to start with a 2 because 40 is even 2 times 20. 20 is even again so I’m going to use 2. 2 times 10, 10 is even so I’m going to use 2 and then 2 times 5 is 10. Now we have 40 broken down into prime factors. These are all prime numbers and 12 is also broken down into prime numbers I’m going to list both of them out 12 is 2 times 2 times 3 and 40 is 2 times 2 times two times five. To get the least common multiple I’m going to use each prime factor once and I’m going to use the largest amount of each prime factor. If you look 12 has two twos but 40 has three twos. I’m going to use these three twos for a least common multiple. I’m going to include 2 times 2 times 2 but I also have to use this 3 because I have to use each factor. I’m going to use a 3 and I have to use this 5. I’m also going to include the 5. To get the least common multiple I’m going to multiply 2 times 2 times 2 times three times five and when I do that, I get 120 as the solution to our least common multiple between 12 and 40.

The next problem on our least common multiple worksheets is number three and that gives us 25 and 80. For this least common multiple example we’re going to do the same thing which is we’re going to take each number and we’re going to break each number down into prime factors. I’m going to take 25 and break it down into 5 times 5. Those are our two prime factors and I know that I’m done because these are both prime numbers. 25 is kind of easy 80 we’re going to break down we’re going to start with 2. 2 times 40 then we’re going to do 2 times 20 to get 40 and then we have to do 20 again. This is 2 times 10 and then 10 is broken down into 2 times five. I’m going to take each number and I’m going to list the prime factors from each number out. We have 25 and we’re going to say 25 is 5 times 5 and I’m going to take 80 and for 80 I’m going to say 80 is 2 times 2 times 2 times 2 times 5. To get the least common multiple that means I have to take each prime factor and the largest amount of each prime factor and multiply them together. In this example we only have two and five so I’m going to use all the twos from 80. This is going to be 2 times 2 times two times two and I’m going to use two fives from 25. This is going to be times five times five now I do not use this five because we already used the two fives from 25. That just sits there we don’t have to use it then to get our least common multiple. I’m going to multiply 2 times 2 times 2 times 2 times 5 times 5 and we will get 400 as the least common multiple between 25 and 80. Hopefully this video is helpful for teaching you how to find the least common multiple. Try all the practice problems by downloading the free finding lcm worksheet above.

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Free Least Common Multiple Worksheet

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