Find the greastest number of 5 digits exactly divisible by 24 , 15 and 36 .

Answer

To find the greatest number of 5 digits that is exactly divisible by 24, 15, and 36, we can follow these steps:

### Step 1: Find the LCM of 24, 15, and 36
To find the least common multiple (LCM), we will first find the prime factorization of each number:

- **24**: \( 24 = 2^3 \times 3^1 \)
- **15**: \( 15 = 3^1 \times 5^1 \)
- **36**: \( 36 = 2^2 \times 3^2 \)

Now, we take 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 36).
- For \(5\): The highest power is \(5^1\) (from 15).

Thus, the LCM is:
\[ \text{LCM} = 2^3 \times 3^2 \times 5^1 = 8 \times 9 \times 5 = 360 \]

### Step 2: Find the greatest 5-digit number
The greatest 5-digit number is \(99999\).

### Step 3: Find the largest multiple of 360 that is less than or equal to 99999
To find this, we divide \(99999\) by \(360\) and take the integer part of the result:
\[ 99999 \div 360 \approx 277.775 \]
Taking the integer part, we get \(277\).

### Step 4: Calculate the greatest multiple of 360
Now, we multiply \(360\) by \(277\):
\[ 360 \times 277 = 99720 \]

### Conclusion
The greatest 5-digit number that is exactly divisible by \(24\), \(15\), and \(36\) is:
\[ \boxed{99720} \]