The Prime Factorizations Of 16 And 24 Are Shown Below.Prime Factorization Of 16: 2, 2, 2, 2 Prime Factorization Of 24: 2, 2, 2, 3 Using The Prime Factorizations, What Is The Greatest Common Factor Of 16 And $24$?A. 2 B. $2 \times

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Introduction

In mathematics, the greatest common factor (GCF) of two numbers is the largest positive integer that divides both numbers without leaving a remainder. In this article, we will explore the prime factorization approach to find the GCF of 16 and 24.

Prime Factorization of 16 and 24

The prime factorization of 16 is 2, 2, 2, 2, which can be written as 242^4. The prime factorization of 24 is 2, 2, 2, 3, which can be written as 23×32^3 \times 3.

Understanding Prime Factorization

Prime factorization is a way of expressing a number as a product of its prime factors. A prime factor is a prime number that divides the original number without leaving a remainder. For example, the prime factorization of 12 is 2, 2, 3, which can be written as 22×32^2 \times 3.

Finding the Greatest Common Factor

To find the GCF of 16 and 24, we need to identify the common prime factors between the two numbers. In this case, the common prime factor is 2. We can see that both 16 and 24 have 2 as a prime factor.

Calculating the Greatest Common Factor

Now that we have identified the common prime factor, we need to calculate the GCF. To do this, we need to find the lowest power of 2 that divides both 16 and 24. In this case, the lowest power of 2 that divides both numbers is 232^3, which is equal to 8.

Conclusion

In conclusion, the greatest common factor of 16 and 24 is 8. This can be verified by listing the factors of 16 and 24 and finding the largest common factor.

Factors of 16

The factors of 16 are 1, 2, 4, 8, 16.

Factors of 24

The factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24.

Common Factors

The common factors of 16 and 24 are 1, 2, 4, 8.

Greatest Common Factor

The greatest common factor of 16 and 24 is 8.

Answer

The correct answer is B. 2×2×2=82 \times 2 \times 2 = 8.

Discussion

This problem can be solved using the prime factorization approach, which involves expressing the numbers as a product of their prime factors. The common prime factor between 16 and 24 is 2, and the lowest power of 2 that divides both numbers is 232^3, which is equal to 8. This can be verified by listing the factors of 16 and 24 and finding the largest common factor.

Related Problems

  • Find the greatest common factor of 18 and 24.
  • Find the greatest common factor of 12 and 18.
  • Find the greatest common factor of 15 and 20.

Solutions

  • The greatest common factor of 18 and 24 is 6.
  • The greatest common factor of 12 and 18 is 6.
  • The greatest common factor of 15 and 20 is 5.

Conclusion

Introduction

In our previous article, we explored the prime factorization approach to find the greatest common factor (GCF) of 16 and 24. In this article, we will answer some frequently asked questions about the GCF and provide additional examples to help you understand this concept better.

Q&A

Q: What is the greatest common factor (GCF)?

A: The greatest common factor (GCF) of two numbers is the largest positive integer that divides both numbers without leaving a remainder.

Q: How do I find the GCF of two numbers?

A: There are several methods to find the GCF, including:

  • Prime factorization: Express the numbers as a product of their prime factors and identify the common prime factors.
  • Listing factors: List the factors of each number and find the largest common factor.
  • Division method: Divide the larger number by the smaller number and find the remainder. Repeat the process until the remainder is 0.

Q: What is the difference between the GCF and the least common multiple (LCM)?

A: The GCF is the largest positive integer that divides both numbers without leaving a remainder, while the LCM is the smallest positive integer that is a multiple of both numbers.

Q: How do I find the LCM of two numbers?

A: To find the LCM, you can use the following formula:

LCM(a, b) = (a × b) / GCF(a, b)

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

A: The GCF has many real-world applications, including:

  • Cooking: When measuring ingredients, you need to find the GCF of the quantities to ensure that you have the correct amount.
  • Building: When building a structure, you need to find the GCF of the dimensions to ensure that the structure is stable.
  • Finance: When investing, you need to find the GCF of the interest rates to ensure that you are getting the best return on your investment.

Q: Can I use a calculator to find the GCF?

A: Yes, you can use a calculator to find the GCF. Most calculators have a built-in function to find the GCF.

Q: What are some common mistakes to avoid when finding the GCF?

A: Some common mistakes to avoid when finding the GCF include:

  • Not listing all the factors: Make sure to list all the factors of each number to ensure that you find the correct GCF.
  • Not identifying the common prime factors: Make sure to identify the common prime factors of each number to ensure that you find the correct GCF.
  • Not using the correct formula: Make sure to use the correct formula to find the LCM.

Examples

Example 1: Find the GCF of 18 and 24

To find the GCF of 18 and 24, we can use the prime factorization method.

18 = 2 × 3 × 3 24 = 2 × 2 × 2 × 3

The common prime factors are 2 and 3. The lowest power of 2 that divides both numbers is 2, and the lowest power of 3 that divides both numbers is 3. Therefore, the GCF of 18 and 24 is 2 × 3 = 6.

Example 2: Find the GCF of 12 and 18

To find the GCF of 12 and 18, we can use the listing factors method.

The factors of 12 are 1, 2, 3, 4, 6, 12. The factors of 18 are 1, 2, 3, 6, 9, 18.

The common factors are 1, 2, 3, 6. Therefore, the GCF of 12 and 18 is 6.

Conclusion

In conclusion, the greatest common factor (GCF) is an important concept in mathematics that has many real-world applications. By understanding the GCF, you can solve problems in cooking, building, finance, and other fields. Remember to use the prime factorization method, listing factors method, or division method to find the GCF, and avoid common mistakes such as not listing all the factors or not identifying the common prime factors.