If the Geometric Mean of two numbers is 16 and their HCF is 8, what is the difference between the LCM and the HCF of the two numbers?

To solve this problem, we need to understand a few concepts:

1. Geometric Mean of two numbers a and b is √(ab).
2. Highest Common Factor (HCF) is the greatest number that divides both numbers.
3. Least Common Multiple (LCM) is the smallest common multiple of both numbers.

Given that the Geometric Mean is 16, we can write:

√(ab) = 16 => ab = 16^2 = 256

Also, we know that the HCF is 8.

In the case of two numbers, the product of the numbers is equal to the product of their HCF and LCM. So we can write:

ab = HCF * LCM => 256 = 8 * LCM => LCM = 256 / 8 = 32

Now, we need to find the difference between the LCM and the HCF:

Difference = LCM - HCF = 32 - 8 = 24

So, the difference between the LCM and the HCF of the two numbers is 24.