Determine The Smallest Number Which Is Exactly Divisible By 6,8 And 12.​

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Introduction

In mathematics, finding the least common multiple (LCM) of a set of numbers is a fundamental concept that has numerous applications in various fields, including algebra, geometry, and number theory. The LCM of a set of numbers is the smallest number that is exactly divisible by each of the numbers in the set. In this article, we will determine the smallest number which is exactly divisible by 6, 8, and 12.

Understanding the Concept of LCM

The LCM of a set of numbers is the product of the highest power of each prime factor that appears in any of the numbers in the set. To find the LCM of a set of numbers, we need to factorize each number into its prime factors and then take the product of the highest power of each prime factor.

Prime Factorization of 6, 8, and 12

To find the LCM of 6, 8, and 12, we need to factorize each number into its prime factors.

  • 6: 6 = 2 × 3
  • 8: 8 = 2^3
  • 12: 12 = 2^2 × 3

Finding the LCM

To find the LCM of 6, 8, and 12, we need to take the product of the highest power of each prime factor that appears in any of the numbers in the set.

  • The highest power of 2 that appears in any of the numbers is 2^3 (from 8).
  • The highest power of 3 that appears in any of the numbers is 3 (from 6 and 12).

Therefore, the LCM of 6, 8, and 12 is:

LCM(6, 8, 12) = 2^3 × 3 = 24

Conclusion

In conclusion, the smallest number which is exactly divisible by 6, 8, and 12 is 24. This is because 24 is the product of the highest power of each prime factor that appears in any of the numbers in the set.

Real-World Applications of LCM

The concept of LCM has numerous real-world applications, including:

  • Music: The LCM of a set of notes is used to determine the frequency of a note.
  • Physics: The LCM of a set of frequencies is used to determine the frequency of a wave.
  • Computer Science: The LCM of a set of numbers is used in algorithms for solving problems such as the greatest common divisor (GCD) and the least common multiple (LCM).

Example Problems

Here are some example problems that involve finding the LCM of a set of numbers:

  • Find the LCM of 4, 6, and 8.
  • Find the LCM of 9, 12, and 15.
  • Find the LCM of 7, 11, and 13.

Solutions to Example Problems

Here are the solutions to the example problems:

  • LCM(4, 6, 8) = 2^3 × 3 = 24
  • LCM(9, 12, 15) = 2^2 × 3^2 × 5 = 180
  • LCM(7, 11, 13) = 7 × 11 × 13 = 1001

Tips and Tricks

Here are some tips and tricks for finding the LCM of a set of numbers:

  • Use prime factorization: Prime factorization is a powerful tool for finding the LCM of a set of numbers.
  • Take the product of the highest power of each prime factor: This is the key to finding the LCM of a set of numbers.
  • Use a calculator or computer program: If you are having trouble finding the LCM of a set of numbers, you can use a calculator or computer program to help you.

Conclusion

Introduction

In our previous article, we discussed how to determine the smallest number which is exactly divisible by 6, 8, and 12. In this article, we will answer some frequently asked questions (FAQs) related to this topic.

Q: What is the least common multiple (LCM) of 6, 8, and 12?

A: The LCM of 6, 8, and 12 is 24.

Q: How do I find the LCM of a set of numbers?

A: To find the LCM of a set of numbers, you need to factorize each number into its prime factors and then take the product of the highest power of each prime factor.

Q: What is prime factorization?

A: Prime factorization is the process of breaking down a number into its prime factors. For example, the prime factorization of 6 is 2 × 3.

Q: How do I find the prime factorization of a number?

A: To find the prime factorization of a number, you can use a factor tree or a calculator.

Q: What is the difference between the LCM and the greatest common divisor (GCD)?

A: The LCM of a set of numbers is the smallest number that is exactly divisible by each of the numbers in the set, while the GCD of a set of numbers is the largest number that divides each of the numbers in the set.

Q: How do I find the GCD of a set of numbers?

A: To find the GCD of a set of numbers, you can use the Euclidean algorithm or a calculator.

Q: What are some real-world applications of the LCM?

A: The LCM has numerous real-world applications, including music, physics, and computer science.

Q: How do I use the LCM in music?

A: In music, the LCM of a set of notes is used to determine the frequency of a note.

Q: How do I use the LCM in physics?

A: In physics, the LCM of a set of frequencies is used to determine the frequency of a wave.

Q: How do I use the LCM in computer science?

A: In computer science, the LCM of a set of numbers is used in algorithms for solving problems such as the greatest common divisor (GCD) and the least common multiple (LCM).

Q: What are some tips and tricks for finding the LCM of a set of numbers?

A: Here are some tips and tricks for finding the LCM of a set of numbers:

  • Use prime factorization: Prime factorization is a powerful tool for finding the LCM of a set of numbers.
  • Take the product of the highest power of each prime factor: This is the key to finding the LCM of a set of numbers.
  • Use a calculator or computer program: If you are having trouble finding the LCM of a set of numbers, you can use a calculator or computer program to help you.

Conclusion

In conclusion, the LCM is a fundamental concept in mathematics that has numerous applications in various fields. We hope that this Q&A article has helped to clarify any questions you may have had about the LCM.

Frequently Asked Questions (FAQs)

Here are some frequently asked questions (FAQs) related to the LCM:

  • Q: What is the LCM of 4, 6, and 8? A: The LCM of 4, 6, and 8 is 24.
  • Q: What is the LCM of 9, 12, and 15? A: The LCM of 9, 12, and 15 is 180.
  • Q: What is the LCM of 7, 11, and 13? A: The LCM of 7, 11, and 13 is 1001.

Additional Resources

Here are some additional resources that you may find helpful:

  • Mathematics textbooks: There are many excellent mathematics textbooks that cover the LCM in detail.
  • Online resources: There are many online resources that provide information and examples related to the LCM.
  • Mathematics software: There are many mathematics software programs that can help you to find the LCM of a set of numbers.