To solve this problem, let's break it down into parts.

Step 1: Find the LCM of 9, 12, and 24

To find the Least Common Multiple (LCM) of 9, 12, and 24, we need to determine the highest power of each prime factor present in the numbers:

1. Prime factorization:

   • $9 = 3^2$
   • $12 = 2^2 \times 3$
   • $24 = 2^3 \times 3$

2. Identify the highest power of each prime factor:

   • For $2$: the highest power is $2^3$ (from 24).
   • For $3$: the highest power is $3^2$ (from 9).

3. Calculate the LCM: $LCM = 2^3 \times 3^2 = 8 \times 9 = 72$

So, $\text{LCM of 9, 12, and 24} = 72$.

Step 2: Find the HCF of 24 and 78

To find the Highest Common Factor (HCF) of 24 and 78, we look for the common prime factors with the lowest powers:

1. Prime factorization:

   • $24 = 2^3 \times 3$
   • $78 = 2 \times 3 \times 13$

2. Identify the common factors:

   • The common factors are $2^1$ and $3^1$.

3. Calculate the HCF: $HCF = 2 \times 3 = 6$

So, $\text{HCF of 24 and 78} = 6$.

Step 3: Divide the LCM by the HCF

Now we can substitute the values we found:

$\frac{\text{LCM of 9, 12, and 24}}{\text{HCF of 24 and 78}} = \frac{72}{6} = 12$

Final Answer

The answer to the problem is 12.

Would you like any further details on any part of this solution?

Here are five related questions that expand on this problem:

1. How do you find the LCM of more than two numbers?
2. What is the difference between LCM and HCF, and when are they used?
3. How can prime factorization help in finding the HCF and LCM?
4. Can you explain the relationship between HCF, LCM, and product of two numbers?
5. How can understanding LCM and HCF help in solving real-life problems?

Tip: Always start by prime factorizing the numbers involved when finding LCM and HCF—it simplifies the process significantly.