Given The Numbers 168 And 320, Answer The Following Questions:5.1 Write The Prime Factors Of 168 And 320.5.2 Determine The LCM (Least Common Multiple) Of 168 And 320.5.3 Determine The GCF (Greatest Common Factor) Of 168 And 320.5.4 Determine The Ratio

Introduction

In mathematics, prime factorization is a process of breaking down a composite number into its prime factors. The Least Common Multiple (LCM) and Greatest Common Factor (GCF) are two important concepts in mathematics that are used to find the smallest multiple and largest factor of two or more numbers. In this article, we will explore the prime factors, LCM, GCF, and ratio of 168 and 320.

Prime Factors of 168 and 320

Prime Factors of 168

To find the prime factors of 168, we need to break it down into its prime factors. The prime factors of 168 are:

• 2 × 2 × 2 × 2 × 3 × 7 = 2^4 × 3 × 7

Prime Factors of 320

To find the prime factors of 320, we need to break it down into its prime factors. The prime factors of 320 are:

• 2 × 2 × 2 × 2 × 2 × 2 × 5 = 2^6 × 5

Determine the LCM (Least Common Multiple) of 168 and 320

The LCM of two numbers is the smallest number that is a multiple of both numbers. To find the LCM of 168 and 320, we need to find the highest power of each prime factor that appears in either number.

• The prime factors of 168 are 2^4, 3, and 7.
• The prime factors of 320 are 2^6 and 5.

The LCM of 168 and 320 is the product of the highest power of each prime factor:

• LCM = 2^6 × 3 × 5 × 7 = 6720

Determine the GCF (Greatest Common Factor) of 168 and 320

The GCF of two numbers is the largest number that divides both numbers. To find the GCF of 168 and 320, we need to find the lowest power of each prime factor that appears in both numbers.

• The prime factors of 168 are 2^4, 3, and 7.
• The prime factors of 320 are 2^6 and 5.

The GCF of 168 and 320 is the product of the lowest power of each prime factor:

• GCF = 2^4 = 16

Determine the Ratio

The ratio of two numbers is the fraction that represents the relationship between the two numbers. To find the ratio of 168 and 320, we can divide 168 by 320:

• Ratio = 168 ÷ 320 = 0.525

However, we can also express the ratio as a fraction:

• Ratio = 168:320 = 21:40

Conclusion

In this article, we have explored the prime factors, LCM, GCF, and ratio of 168 and 320. We have found that the prime factors of 168 are 2^4, 3, and 7, and the prime factors of 320 are 2^6 and 5. We have also found that the LCM of 168 and 320 is 6720, the GCF is 16, and the ratio is 21:40. These concepts are important in mathematics and have many practical applications in real-world problems.

References

• [1] Khan Academy. (n.d.). Prime Factorization. Retrieved from https://www.khanacademy.org/math/algebra/x2factors/x2prime_factors
• [2] Math Open Reference. (n.d.). Least Common Multiple (LCM). Retrieved from https://www.mathopenref.com/lcm.html
• [3] Math Open Reference. (n.d.). Greatest Common Factor (GCF). Retrieved from https://www.mathopenref.com/gcf.html

Further Reading

• [1] Prime Factorization: A Guide to Factoring Numbers. (n.d.). Retrieved from https://www.mathsisfun.com/prime-factorization.html
• [2] Least Common Multiple (LCM) and Greatest Common Factor (GCF). (n.d.). Retrieved from https://www.mathsisfun.com/least-common-multiple-greatest-common-factor.html
  Prime Factors, LCM, GCF, and Ratio Q&A =============================================

Q: What is prime factorization?

A: Prime factorization is the process of breaking down a composite number into its prime factors. A prime factor is a prime number that can be multiplied together to get the original number.

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

A: To find the prime factors of a number, you can use the following steps:

1. Start by dividing the number by the smallest prime number, which is 2.
2. If the number is divisible by 2, then 2 is a prime factor.
3. Continue dividing the number by 2 until it is no longer divisible.
4. Then, move on to the next prime number, which is 3.
5. Repeat the process of dividing the number by 3 until it is no longer divisible.
6. Continue this process with the next prime numbers, which are 5, 7, 11, and so on.

Q: What is the Least Common Multiple (LCM)?

A: The LCM of two numbers is the smallest number that is a multiple of both numbers. To find the LCM, you can list the multiples of each number and find the smallest multiple that appears in both lists.

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

A: To find the LCM of two numbers, you can use the following steps:

1. List the multiples of each number.
2. Find the smallest multiple that appears in both lists.
3. The LCM is the smallest multiple that appears in both lists.

Q: What is the Greatest Common Factor (GCF)?

A: The GCF of two numbers is the largest number that divides both numbers. To find the GCF, you can list the factors of each number and find the largest factor that appears in both lists.

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

A: To find the GCF of two numbers, you can use the following steps:

1. List the factors of each number.
2. Find the largest factor that appears in both lists.
3. The GCF is the largest factor that appears in both lists.

Q: What is the ratio of two numbers?

A: The ratio of two numbers is the fraction that represents the relationship between the two numbers. To find the ratio, you can divide the first number by the second number.

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

A: To find the ratio of two numbers, you can use the following steps:

1. Divide the first number by the second number.
2. Simplify the fraction to its simplest form.

Q: What is the difference between the LCM and GCF?

A: The LCM and GCF are two different concepts in mathematics. The LCM is the smallest number that is a multiple of both numbers, while the GCF is the largest number that divides both numbers.

Q: How do I use the LCM and GCF in real-world problems?

A: The LCM and GCF are used in many real-world problems, such as:

• Finding the smallest multiple of two numbers
• Finding the largest factor of two numbers
• Converting between different units of measurement
• Solving problems involving fractions and decimals

Q: What are some common mistakes to avoid when working with the LCM and GCF?

A: Some common mistakes to avoid when working with the LCM and GCF include:

• Not simplifying the fraction to its simplest form
• Not finding the smallest multiple or largest factor
• Not using the correct formula or method
• Not checking the work for errors

Q: How can I practice working with the LCM and GCF?

A: You can practice working with the LCM and GCF by:

• Using online resources and worksheets
• Working with real-world problems and examples
• Practicing with different numbers and scenarios
• Asking a teacher or tutor for help and guidance

Conclusion

In this article, we have explored the prime factors, LCM, GCF, and ratio of two numbers. We have also answered some common questions and provided tips and resources for practicing and mastering these concepts. With practice and patience, you can become proficient in working with the LCM and GCF and apply these concepts to real-world problems.