Finding HCF and LCM of 98 and 28

Oct 13, 2025 by miodeya.com 94 views

Introduction

In mathematics, the concept of Highest Common Factor (HCF) and Least Common Multiple (LCM) is crucial for solving various problems related to numbers. The HCF of two numbers is the greatest number that divides both of them without leaving a remainder, while the LCM is the smallest number that is a multiple of both numbers. In this article, we will explore the relationship between HCF and LCM, and how to use it to find the value of n - 7m, where m is the HCF of 98 and 28, and n is the LCM of 98 and 28.

Understanding HCF and LCM

To begin with, let's understand the concept of HCF and LCM. The HCF of two numbers is the product of the common prime factors of the two numbers, raised to the power of the smallest exponent. On the other hand, the LCM of two numbers is the product of the highest power of all the prime factors of the two numbers.

For example, let's consider the numbers 12 and 18. The prime factorization of 12 is 2^2 * 3, and the prime factorization of 18 is 2 * 3^2. The HCF of 12 and 18 is 2 * 3 = 6, which is the product of the common prime factors of the two numbers, raised to the power of the smallest exponent. The LCM of 12 and 18 is 2^2 * 3^2 = 36, which is the product of the highest power of all the prime factors of the two numbers.

Finding HCF and LCM of 98 and 28

Now, let's find the HCF and LCM of 98 and 28. The prime factorization of 98 is 2 * 7^2, and the prime factorization of 28 is 2^2 * 7. The HCF of 98 and 28 is 2 * 7 = 14, which is the product of the common prime factors of the two numbers, raised to the power of the smallest exponent. The LCM of 98 and 28 is 2^2 * 7^2 = 196, which is the product of the highest power of all the prime factors of the two numbers.

Finding the Value of n - 7m

Now that we have found the HCF and LCM of 98 and 28, we can find the value of n - 7m. We know that m is the HCF of 98 and 28, which is 14, and n is the LCM of 98 and 28, which is 196. Therefore, the value of n - 7m is 196 - 7(14) = 196 - 98 = 98.

Conclusion

In conclusion, we have found the value of n - 7m, where m is the HCF of 98 and 28, and n is the LCM of 98 and 28. The value of n - 7m is 98, which is the difference between the LCM and 7 times the HCF of the two numbers.

Frequently Asked Questions

What is the HCF of 98 and 28? The HCF of 98 and 28 is 14.
What is the LCM of 98 and 28? The LCM of 98 and 28 is 196.
What is the value of n - 7m? The value of n - 7m is 98.

Step-by-Step Solution

Find the prime factorization of 98 and 28.
Find the HCF of 98 and 28 by taking the product of the common prime factors, raised to the power of the smallest exponent.
Find the LCM of 98 and 28 by taking the product of the highest power of all the prime factors.
Find the value of n - 7m by subtracting 7 times the HCF from the LCM.

Tips and Tricks

To find the HCF and LCM of two numbers, you can use the prime factorization method.
The HCF of two numbers is the product of the common prime factors, raised to the power of the smallest exponent.
The LCM of two numbers is the product of the highest power of all the prime factors.
To find the value of n - 7m, you can subtract 7 times the HCF from the LCM.

Q&A: Understanding HCF and LCM

Q: What is the Highest Common Factor (HCF) of two numbers?

A: The HCF of two numbers is the greatest number that divides both of them without leaving a remainder.

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

A: The LCM of two numbers is the smallest number that is a multiple of both numbers.

Q: How do you find the HCF and LCM of two numbers?

A: To find the HCF and LCM of two numbers, you can use the prime factorization method. The HCF is the product of the common prime factors, raised to the power of the smallest exponent, while the LCM is the product of the highest power of all the prime factors.

Q: What is the relationship between HCF and LCM?

A: The relationship between HCF and LCM is given by the formula: HCF(a,b) * LCM(a,b) = a * b, where a and b are the two numbers.

Q: How do you find the value of n - 7m?

A: To find the value of n - 7m, you can subtract 7 times the HCF from the LCM. In this case, the value of n - 7m is 196 - 7(14) = 196 - 98 = 98.

Q&A: Finding HCF and LCM of 98 and 28

Q: What is the prime factorization of 98?

A: The prime factorization of 98 is 2 * 7^2.

Q: What is the prime factorization of 28?

A: The prime factorization of 28 is 2^2 * 7.

Q: What is the HCF of 98 and 28?

A: The HCF of 98 and 28 is 2 * 7 = 14.

Q: What is the LCM of 98 and 28?

A: The LCM of 98 and 28 is 2^2 * 7^2 = 196.

Q&A: Tips and Tricks

Q: What is the best way to find the HCF and LCM of two numbers?

A: The best way to find the HCF and LCM of two numbers is to use the prime factorization method.

Q: How do you ensure that you have found the correct HCF and LCM?

A: To ensure that you have found the correct HCF and LCM, you can check that the product of the HCF and LCM is equal to the product of the two numbers.

Q: What are some common mistakes to avoid when finding the HCF and LCM?

A: Some common mistakes to avoid when finding the HCF and LCM include not using the prime factorization method, not checking that the product of the HCF and LCM is equal to the product of the two numbers, and not being careful when calculating the HCF and LCM.

Q&A: Conclusion

Q: What is the value of n - 7m?

A: The value of n - 7m is 98.

Q: What is the relationship between HCF and LCM?

A: The relationship between HCF and LCM is given by the formula: HCF(a,b) * LCM(a,b) = a * b, where a and b are the two numbers.