What is the greatest 5-digit number which is exactly divisible by 36, 42, and 48?

Let's solve the problem step-by-step:

### Step 1: Calculate the Least Common Multiple (LCM) of 36, 42, and 48.

First, find the prime factorizations:
- \( 36 = 2^2 \times 3^2 \)
- \( 42 = 2 \times 3 \times 7 \)
- \( 48 = 2^4 \times 3 \)

To calculate the LCM, take the highest powers of all the prime factors involved:
- \( 2^4 \) (from 48)
- \( 3^2 \) (from 36)
- \( 7 \) (from 42)

So, the LCM is:
\[
LCM = 2^4 \times 3^2 \times 7 = 16 \times 9 \times 7 = 1008
\]

### Step 2: Determine the greatest 5-digit number divisible by 1008.

The largest 5-digit number is 99999. We now divide 99999 by 1008 to find the closest multiple:
\[
\frac{99999}{1008} \approx 99.21
\]

The largest integer quotient is 99.

### Step 3: Multiply the quotient by 1008 to find the greatest 5-digit multiple.
\[
99 \times 1008 = 99792
\]

### Conclusion:
The greatest 5-digit number exactly divisible by 36, 42, and 48 is **99792**.

Would you like more details or have any questions?

Here are 5 related questions to explore:
1. How can you calculate the LCM using the division method?
2. What role does prime factorization play in finding the LCM?
3. How would the approach change if you had different numbers?
4. What if you needed the smallest 5-digit number divisible by 36, 42, and 48?
5. Can you explain why 99792 is the correct answer using a different method?

**Tip:** When solving problems involving multiple divisibility conditions, always start by finding the LCM to ensure all conditions are satisfied.

Learn how to find the greatest 5-digit number exactly divisible by 36, 42, and 48. Step-by-step solution using LCM calculation and integer division. Suitable for high school students.

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