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

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 \

Discover how to find the greatest 5-digit number which is exactly divisible by 36, 42, and 48. Learn the steps involving number theory and LCM calculation, suitable for students in Grades 7-9.

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