The H.C.F. and L.C.M. Relationship: A Step-by-Step Guide

In mathematics, particularly in number theory, understanding the relationship between the Highest Common Factor (H.C.F.) and the Lowest Common Multiple (L.C.M.) is crucial for solving a variety of problems. This article explains how to find the other number when given the H.C.F., L.C.M., and one of the numbers. We will demonstrate this with a practical example and explore the underlying mathematical principles.

Understanding the Key Concepts

Before solving the problem, it is essential to understand the definitions of H.C.F. and L.C.M.

What is H.C.F.?

The H.C.F., or Highest Common Factor, is the largest positive integer that divides both numbers without leaving a remainder. It is also known as the Greatest Common Divisor (G.C.D.).

What is L.C.M.?

The L.C.M., or Lowest Common Multiple, is the smallest positive integer that is a multiple of both numbers. It represents the smallest common denominator when adding or subtracting fractions with different denominators.

Example Problem

Given that the H.C.F. of two numbers is 11 and their L.C.M. is 693. If one of the numbers is 77, what is the other number?

Step-by-Step Solution

Let's denote the two numbers as (n_1) and (n_2). Given:

(H.C.F. (n_1, n_2) 11) (L.C.M. (n_1, n_2) 693) (n_1 77)

Using the relationship between H.C.F. and L.C.M., we have:

H.C.F. (times L.C.M. n_1 times n_2)

Substituting the given values:

11 (times 693 77 times n_2)

Calculating the left side:

11 (times 693 7623)

Setting up the equation:

7623 77 times n_2)

Solving for (n_2), we divide both sides by 77:

n_2 frac{7623}{77})

Calculating the division:

n_2 99

Therefore, the other number is 99.

Conclusion

By using the relationship between the H.C.F. and L.C.M., we can find the missing number. This method is consistent and reliable for solving problems involving these concepts. The other number, in this case, is 99.

Related Questions

1. If the H.C.F. of two numbers is 22 and their L.C.M. is 693, and one of the numbers is 77, what is the other number?

2. Given that H.C.F. is 11 and L.C.M. is 368, and the product of the two numbers is 22693, find the other number.

Additional Resources

For further reading and practice, explore resources on number theory, such as:

MathIsFun - H.C.F. and L.C.M. Calculators Khan Academy - Factors and Multiples Cuemath - L.C.M. and H.C.F. Problems

These resources are great for strengthening your understanding of number theory and improving problem-solving skills.