Understanding the Relationship Between Product, HCF, and LCM

In mathematical terms, the Least Common Multiple (LCM) and the Highest Common Factor (HCF) are crucial concepts in number theory. When given the product of two numbers and their HCF, the LCM can be determined using a straightforward formula. This article provides a detailed explanation of this relationship, along with a step-by-step solution to a specific problem.

The Relationship Formula

The relationship between the product of two numbers, their HCF (Highest Common Factor), and their LCM (Least Common Multiple) is given by the formula: [ text{Product of the two numbers} text{HCF} times text{LCM} ] This relationship is derived from the fundamental properties of divisors and multiples in number theory.

Solving a Specific Problem

Consider the problem where the product of two numbers is 1080 and their HCF is 30. We are tasked with finding the LCM of these two numbers. The problem can be broken down as follows:

We start by understanding that the HCF of the two numbers is 30. This means each number can be written as a multiple of 30. The product of the two numbers is given as 1080. The formula to find the LCM is:

[ text{LCM} frac{text{Product of the two numbers}}{text{HCF}} ]

Applying the given values to the formula, we substitute the product and the HCF:

[ text{LCM} frac{1080}{30} 36 ]

Thus, the LCM of the two numbers is 36.

Theoretical Approach

Another way to solve this problem is by using the theoretical approach. Let the two numbers be 30a and 30b, where a and b are coprime numbers (i.e., their HCF is 1). This means that the product of the two numbers can be written as:

[ 30a times 30b 900ab ]

Given that the product is 1080, we can set up the equation:

[ 900ab 1080 ]

Simplifying the above equation, we get:

[ ab frac{1080}{900} 1.2 ]

Since a and b are coprime and their product must be an integer, no such pair of natural numbers exists that satisfies this condition. However, theoretically, the relationship still holds:

[ text{HCF} times text{LCM} text{Product of the two numbers} ]

Substituting the given values, we get:

[ 30 times text{LCM} 1080 ]

Solving for the LCM, we find:

[ text{LCM} frac{1080}{30} 36 ]

Thus, the LCM is 36.

Conclusion

Understanding the relationship between the product of two numbers, their HCF, and their LCM is crucial in solving various mathematical problems. By using the given formula and applying theoretical knowledge, we can determine the LCM accurately. This method is widely applicable in various mathematical and practical scenarios.

The key formulas to remember are:

[ text{Product of the two numbers} text{HCF} times text{LCM} ] [ text{LCM} frac{text{Product of the two numbers}}{text{HCF}} ]

With these formulas, you can easily solve similar problems involving the LCM, HCF, and product of two numbers.