Expressing the Final Answer in Exponential Form

Oct 13, 2025 by windailys.com 88 views

Introduction

In mathematics, prime factorization and lowest common multiple (LCM) are two fundamental concepts that play a crucial role in solving various mathematical problems. Prime factorization involves expressing a number as a product of its prime factors, while LCM is the smallest multiple that is common to a set of numbers. In this article, we will explore these concepts in detail and provide step-by-step solutions to two problems: writing 2457 as a product of prime factors and determining the LCM of 75 and 100.

Prime Factorization

Prime factorization is the process of expressing a number as a product of its prime factors. A prime factor is a prime number that divides the given number exactly. For example, the prime factorization of 12 is 2 × 2 × 3, since 2 and 3 are prime numbers that divide 12 exactly.

Step 1: Write 2457 as a Product of Prime Factors

To write 2457 as a product of prime factors, we need to find the prime factors of 2457. We can do this by dividing 2457 by the smallest prime number, which is 2. If 2457 is not divisible by 2, we move on to the next prime number, which is 3.

2457 ÷ 3 = 819
819 ÷ 3 = 273
273 ÷ 3 = 91
91 ÷ 7 = 13

Therefore, the prime factorization of 2457 is 3 × 3 × 3 × 7 × 13.

Expressing the Final Answer in Exponential Form

We can express the final answer in exponential form by writing the prime factors with their respective exponents. In this case, the prime factorization of 2457 can be written as 3^3 × 7 × 13.

Lowest Common Multiple (LCM)

The LCM of a set of numbers is the smallest multiple that is common to all the numbers in the set. To find the LCM of two numbers, we need to find the prime factorization of each number and then take the highest power of each prime factor that appears in either number.

Step 2: Determine the LCM of 75 and 100

To determine the LCM of 75 and 100, we need to find the prime factorization of each number.

75 = 3 × 5 × 5
100 = 2 × 2 × 5 × 5

The LCM of 75 and 100 is the product of the highest power of each prime factor that appears in either number. In this case, the LCM is 2 × 2 × 3 × 5 × 5 = 300.

Conclusion

In conclusion, prime factorization and LCM are two fundamental concepts in mathematics that play a crucial role in solving various mathematical problems. By understanding these concepts, we can express numbers as products of their prime factors and find the smallest multiple that is common to a set of numbers. In this article, we provided step-by-step solutions to two problems: writing 2457 as a product of prime factors and determining the LCM of 75 and 100.

References

[1] Khan Academy. (n.d.). Prime Factorization. Retrieved from https://www.khanacademy.org/math/algebra/x2factors/x2factors-prime-factorization
[2] Math Open Reference. (n.d.). Prime Factorization. Retrieved from https://www.mathopenref.com/primefactorization.html

Frequently Asked Questions

Q: What is prime factorization? A: Prime factorization is the process of expressing a number as a product of its prime factors.
Q: How do I find the prime factorization of a number? A: To find the prime factorization of a number, you need to divide the number by the smallest prime number, which is 2. If the number is not divisible by 2, you move on to the next prime number, which is 3.
Q: What is the LCM of two numbers? A: The LCM of two numbers is the smallest multiple that is common to both numbers.
Q: How do I find the LCM of two numbers? A: To find the LCM of two numbers, you need to find the prime factorization of each number and then take the highest power of each prime factor that appears in either number.
Prime Factorization and Lowest Common Multiple: A Comprehensive Guide ===========================================================

Q&A: Prime Factorization and Lowest Common Multiple

Q: What is prime factorization?

A: Prime factorization is the process of expressing a number as a product of its prime factors. A prime factor is a prime number that divides the given number exactly.

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

A: To find the prime factorization of a number, you need to divide the number by the smallest prime number, which is 2. If the number is not divisible by 2, you move on to the next prime number, which is 3. You continue this process until you have divided the number by all prime numbers.

Q: What is the lowest common multiple (LCM)?

A: The LCM of a set of numbers is the smallest multiple that is common to all the numbers in the set.

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

A: To find the LCM of two numbers, you need to find the prime factorization of each number and then take the highest power of each prime factor that appears in either number.

Q: What is the difference between prime factorization and LCM?

A: Prime factorization involves expressing a number as a product of its prime factors, while LCM involves finding the smallest multiple that is common to a set of numbers.

Q: Can you give an example of prime factorization?

A: Yes, the prime factorization of 12 is 2 × 2 × 3, since 2 and 3 are prime numbers that divide 12 exactly.

Q: Can you give an example of LCM?

A: Yes, the LCM of 6 and 8 is 24, since 24 is the smallest multiple that is common to both 6 and 8.

Q: How do I use prime factorization to find the LCM of two numbers?

A: To use prime factorization to find the LCM of two numbers, you need to find the prime factorization of each number and then take the highest power of each prime factor that appears in either number.

Q: What is the importance of prime factorization and LCM in real-life situations?

A: Prime factorization and LCM are important in real-life situations such as cryptography, coding theory, and number theory. They are also used in various fields such as engineering, physics, and computer science.

Q: Can you give some examples of real-life applications of prime factorization and LCM?

A: Yes, some examples of real-life applications of prime factorization and LCM include:

Cryptography: Prime factorization is used to break certain types of encryption.
Coding theory: Prime factorization is used to construct error-correcting codes.
Number theory: Prime factorization and LCM are used to study the properties of integers.
Engineering: Prime factorization and LCM are used to design and analyze electrical circuits.
Physics: Prime factorization and LCM are used to study the properties of particles and forces.

Q: How do I practice prime factorization and LCM?

A: You can practice prime factorization and LCM by solving problems and exercises. You can also use online resources and tools to help you practice.

Q: What are some common mistakes to avoid when working with prime factorization and LCM?

A: Some common mistakes to avoid when working with prime factorization and LCM include:

Not checking if a number is prime before dividing it.
Not taking the highest power of each prime factor.
Not using the correct method to find the LCM.

Q: Can you give some tips for mastering prime factorization and LCM?

A: Yes, some tips for mastering prime factorization and LCM include:

Practice regularly to build your skills and confidence.
Use online resources and tools to help you practice.
Review and practice regularly to reinforce your understanding.
Seek help from a teacher or tutor if you are struggling.

Conclusion

In conclusion, prime factorization and LCM are two fundamental concepts in mathematics that play a crucial role in solving various mathematical problems. By understanding these concepts, you can express numbers as products of their prime factors and find the smallest multiple that is common to a set of numbers. We hope that this article has provided you with a comprehensive guide to prime factorization and LCM, and that you will find it helpful in your studies.