Understanding the Relationship Between the Highest Common Factor (HCF), Lowest Common Multiple (LCM), and the Product of Two Numbers

The relationship between the HCF and LCM of two numbers and their product is a fundamental concept in mathematics, often used in solving complex problems. This relationship can be particularly useful in areas such as number theory, algebra, and even in practical applications like scheduling and resource allocation. Let's explore this concept through an example.

Example Problem

Given: The HCF of two numbers is 12, and their LCM is 72. What is the product of these two numbers?

Step 1: Understanding the Relationship

The relationship between the HCF, LCM, and the product of two numbers is given by the formula:

Product of two numbers HCF × LCM

Step 2: Substituting the Given Values

Given the HCF (Highest Common Factor) is 12 and the LCM (Lowest Common Multiple) is 72, we can substitute these values into the formula:

Product of two numbers 12 × 72

Step 3: Calculating the Product

Now, perform the multiplication:

Product of two numbers 864

Conclusion

Therefore, the product of the two numbers is 864.

Further Explorations

Let's further explore this concept using a different pair of numbers to illustrate the consistency of the formula:

Example 1

Consider two numbers, 15 and 35. First, we determine their HCF and LCM:

15 3 × 5

35 5 × 7

HCF(15, 35) 5 (since 5 is the highest factor common to both numbers)

LCM(15, 35) 3 × 5 × 7 105 (since the LCM is the smallest multiple common to both numbers)

Using the formula:

Product of 15 and 35 HCF × LCM 5 × 105 525

This matches the direct product of 15 and 35:

15 × 35 525

Example 2

Consider two other numbers, 12 and 72. We should verify if the same relationship holds:

HCF(12, 72) 12 (since 12 is the highest factor common to both numbers)

LCM(12, 72) 72 (since 72 is the smallest multiple common to both numbers)

Using the formula:

Product of 12 and 72 HCF × LCM 12 × 72 864

This matches the direct product of 12 and 72:

12 × 72 864

Conceptual Explanation

The formula can be derived from the prime factorization of the numbers. If we express the two numbers as multiples of their HCF, we get:

Number 1 HCF × x

Number 2 HCF × y

Where x and y are relatively prime (they have no common factors other than 1).

The LCM is then given by:

LCM(Number 1, Number 2) HCF × x × y

The product of the two numbers is:

Number 1 × Number 2 (HCF × x) × (HCF × y) HCF × HCF × x × y

Since LCM HCF × x × y, the product can be written as:

Product HCF × LCM

Conclusion

The formula to find the product of two numbers using their HCF and LCM is a powerful tool in mathematics. It simplifies many problems and can be applied in various mathematical and practical scenarios. By understanding the interplay between HCF, LCM, and the product of two numbers, we can solve complex problems more efficiently.