The Product of LCM and HCF of Two Numbers is 80. If One Number is 20, What is the Other Number?

When dealing with number theory problems involving the Least Common Multiple (LCM) and the Highest Common Factor (HCF), also known as the Greatest Common Divisor (GCD), we can use the relationship between these properties to solve for unknown numbers. The key relationship is:

LCM × HCF Number 1 × Number 2

In this problem, we are given that the product of the LCM and HCF of two numbers is 80, and one of the numbers is 20. We need to find the other number.

Given Information and Formulation

Let's denote the two numbers as N1 and N2, where N1 20 and the product of the LCM and HCF is 80.

The relationship can be written as:

LCM × HCF N1 × N2

Setting Up an Equation

Given:

tThe product of LCM and HCF 80 tOne number, N1 20

We need to find the other number, N2.

Using the relationship and the given data, we can set up the equation as:

80 20 × N2

Solving for the Other Number

To solve for N2, we divide both sides of the equation by 20:

N2 80 / 20

N2 4

Therefore, the other number is 4.

Verification

To verify our solution, we can check the HCF and LCM of 4 and 20.

tFactors of 4 2 × 2 tFactors of 20 2 × 2 × 5 tThe Highest Common Factor (HCF/GCD) is 2 × 2 4 tThe Least Common Multiple (LCM) is 2 × 2 × 5 20

The product of 4 and 20 is:

4 × 20 80

This confirms that our solution is correct.

Summary

In this article, we have demonstrated how to solve for an unknown number using the relationship between the product of the LCM and HCF (or GCD) of two numbers. Given the product of LCM and HCF is 80 and one number is 20, we found the other number to be 4. This problem reinforces the importance of understanding number theory and its applications in solving equations.

Key Takeaways

tThe relationship LCM × HCF Number 1 × Number 2 is a fundamental property in number theory. tHCF (or GCD) and LCM can be used to solve for unknown numbers. tVerification through factorization helps confirm the correctness of the solution.