Find maximum number of 3 digit exactly divisible by 2,3,9,11.

Answer

To find the maximum number of 3-digit numbers that are exactly divisible by 2, 3, 9, and 11, we will follow these steps:

### Step 1: Find the LCM of the given numbers
To determine the numbers that are divisible by 2, 3, 9, and 11, we first need to calculate the Least Common Multiple (LCM) of these numbers.

- The prime factorization of the numbers is as follows:
- 2 = 2
- 3 = 3
- 9 = 3^2
- 11 = 11

- The LCM is found by taking the highest power of each prime factor:
- LCM = 2^1 * 3^2 * 11^1 = 2 * 9 * 11 = 198

### Step 2: Identify the range of 3-digit numbers
The range of 3-digit numbers is from 100 to 999.

### Step 3: Find the largest 3-digit number divisible by the LCM
To find the largest 3-digit number that is divisible by 198, we divide 999 by 198 and find the largest integer less than or equal to the result.

- Calculate:
\[ 999 \div 198 \approx 5.0505 \]

- The largest integer less than or equal to 5.0505 is 5.

### Step 4: Calculate the largest 3-digit number divisible by 198
Now, we multiply the integer we found by 198 to get the largest 3-digit number divisible by 198.

- Calculate:
\[ 5 \times 198 = 990 \]

### Step 5: Count the number of 3-digit numbers divisible by 198
Next, we need to find how many multiples of 198 are there in the 3-digit range (from 100 to 999).

- The smallest 3-digit number divisible by 198 can be found by taking the ceiling of the division of 100 by 198:
\[ 100 \div 198 \approx 0.5051 \quad \text{(round up to 1)} \]

- Now, the smallest 3-digit number divisible by 198:
\[ 1 \times 198 = 198 \]

### Step 6: Count the multiples of 198 from 198 to 990
Now we can find how many multiples of 198 exist between 198 and 990.

- The multiples of 198 can be expressed as:
\[ 198, 396, 594, 792, 990 \]

- To count them, we can use the formula:
\[ n = \frac{\text{largest multiple} - \text{smallest multiple}}{\text{LCM}} + 1 \]
\[ n = \frac{990 - 198}{198} + 1 = \frac{792}{198} + 1 = 4 + 1 = 5 \]

### Conclusion
Thus, the maximum number of 3-digit numbers that are exactly divisible by 2, 3, 9, and 11 is **5**.

---