To determine the greatest 5-digit number exactly divisible by 36, 42, and 48, we follow these steps:

Step 1: Calculate the Least Common Multiple (LCM) of the numbers.

We find the LCM of 36, 42, and 48.

First, find the prime factorizations:

• $36 = 2^2 \times 3^2$
• $42 = 2 \times 3 \times 7$
• $48 = 2^4 \times 3$

Now, take the highest powers of all the prime factors involved:

• $2^4$ (from 48)
• $3^2$ (from 36)
• $7$ (from 42)

Thus, the LCM is: $LCM = 2^4 \times 3^2 \times 7 = 16 \times 9 \times 7 = 1008$

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

The largest 5-digit number is 99999. We now divide it by 1008 to find the closest multiple:

$\text{Quotient} = \left\lfloor \frac{99999}{1008} \right\rfloor = 99$

Multiplying this quotient back by 1008 gives: [ 99 \