Using LCM and HCF to Find the Other Number

Oct 13, 2025 by miodeya.com 97 views

Introduction

In mathematics, the Least Common Multiple (LCM) and the Highest Common Factor (HCF) are two fundamental concepts that are used to find the relationship between two or more numbers. The LCM of two numbers is the smallest number that is a multiple of both, while the HCF is the greatest number that divides both numbers without leaving a remainder. In this article, we will explore how to use the LCM and HCF to find the other number when one of the numbers is given.

Understanding LCM and HCF

The LCM of two numbers is the product of the two numbers divided by their HCF. This can be expressed mathematically as:

LCM(a, b) = (a × b) / HCF(a, b)

where a and b are the two numbers.

The HCF of two numbers is the greatest number that divides both numbers without leaving a remainder. It can be found using the prime factorization method or the Euclidean algorithm.

Given Information

We are given that the LCM of the two numbers is 360 and their HCF is 24. We are also given that one of the numbers is 120. We need to find the other number.

Using LCM and HCF to Find the Other Number

We can use the formula for LCM to find the other number. Since we know the LCM and the HCF, we can rearrange the formula to get:

a × b = LCM(a, b) × HCF(a, b)

Substituting the given values, we get:

a × 120 = 360 × 24

Simplifying the equation, we get:

a × 120 = 8640

Now, we can divide both sides by 120 to get:

a = 8640 / 120

a = 72

Therefore, the other number is 72.

Verification

To verify our answer, we can find the LCM and HCF of 120 and 72.

The prime factorization of 120 is:

120 = 2^3 × 3 × 5

The prime factorization of 72 is:

72 = 2^3 × 3^2

The HCF of 120 and 72 is the product of the common prime factors:

HCF(120, 72) = 2^3 × 3 = 24

The LCM of 120 and 72 is the product of the two numbers divided by their HCF:

LCM(120, 72) = (120 × 72) / 24

LCM(120, 72) = 360

Therefore, our answer is correct.

Conclusion

In this article, we used the LCM and HCF to find the other number when one of the numbers is given. We found that the other number is 72. We also verified our answer by finding the LCM and HCF of 120 and 72. This demonstrates the power of using LCM and HCF in solving mathematical problems.

Real-World Applications

The concept of LCM and HCF has many real-world applications. For example, in music, the LCM of two notes is used to find the frequency of the notes. In engineering, the HCF of two materials is used to find the strength of the materials. In finance, the LCM of two investments is used to find the return on investment.

Frequently Asked Questions

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

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

Q: How do I find the LCM and HCF of two numbers? A: You can use the prime factorization method or the Euclidean algorithm to find the LCM and HCF of two numbers.

Q: What is the relationship between LCM and HCF? A: The LCM of two numbers is the product of the two numbers divided by their HCF.

References

"Mathematics for Dummies" by Mark Ryan
"Algebra and Trigonometry" by James Stewart
"Calculus" by Michael Spivak

Glossary

LCM: Least Common Multiple
HCF: Highest Common Factor
Prime factorization: The process of finding the prime factors of a number.
Euclidean algorithm: A method for finding the greatest common divisor of two numbers.
The LCM and HCF of Two Numbers: A Q&A Guide =====================================================

Introduction

In our previous article, we explored how to use the Least Common Multiple (LCM) and the Highest Common Factor (HCF) to find the other number when one of the numbers is given. In this article, we will provide a Q&A guide to help you understand the concept of LCM and HCF and how to apply it in different situations.

Q&A Guide

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

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

Q: How do I find the LCM and HCF of two numbers? A: You can use the prime factorization method or the Euclidean algorithm to find the LCM and HCF of two numbers.

Q: What is the relationship between LCM and HCF? A: The LCM of two numbers is the product of the two numbers divided by their HCF.

Q: How do I use LCM and HCF to find the other number when one of the numbers is given? A: You can use the formula: a × b = LCM(a, b) × HCF(a, b) to find the other number.

Q: What is the difference between LCM and HCF? A: The LCM of two numbers is the smallest number that is a multiple of both, while the HCF is the greatest number that divides both numbers without leaving a remainder.

Q: Can I use LCM and HCF to find the prime factors of a number? A: Yes, you can use LCM and HCF to find the prime factors of a number. For example, if the LCM of two numbers is 360 and their HCF is 24, you can find the prime factors of 360 and 24 to determine the prime factors of the two numbers.

Q: How do I use LCM and HCF in real-world applications? A: You can use LCM and HCF in various real-world applications such as music, engineering, and finance. For example, in music, the LCM of two notes is used to find the frequency of the notes, while in engineering, the HCF of two materials is used to find the strength of the materials.

Q: What are some common mistakes to avoid when using LCM and HCF? A: Some common mistakes to avoid when using LCM and HCF include:

Not using the correct formula to find the LCM and HCF
Not considering the prime factors of the numbers
Not using the correct method to find the LCM and HCF
Not verifying the results

Q: How do I verify the results when using LCM and HCF? A: You can verify the results by checking the prime factors of the numbers and ensuring that the LCM and HCF are correct.

Q: What are some advanced topics related to LCM and HCF? A: Some advanced topics related to LCM and HCF include:

Using LCM and HCF to find the greatest common divisor (GCD) of two numbers
Using LCM and HCF to find the least common multiple (LCM) of three or more numbers
Using LCM and HCF to find the prime factors of a number