Find the greatest number of 4 digits which is exactly divisible by 15 , 24 and 36.

Answer

To find the greatest four-digit number that is exactly divisible by 15, 24, and 36, we will follow these steps:

### Step 1: Find the LCM of 15, 24, and 36
To find the least common multiple (LCM), we first need to factor each number into its prime factors:

- **15**: \(3 \times 5\)
- **24**: \(2^3 \times 3\)
- **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 calculated as:
\[ \text{LCM} = 2^3 \times 3^2 \times 5^1 = 8 \times 9 \times 5 \]
Calculating this step-by-step:
\[ 8 \times 9 = 72 \]
\[ 72 \times 5 = 360 \]
So, the LCM of 15, 24, and 36 is **360**.

### Step 2: Find the greatest four-digit number
The greatest four-digit number is **9999**.

### Step 3: Check divisibility by 360
Now, we need to find the largest four-digit number that is divisible by 360. We do this by dividing 9999 by 360 and finding the largest integer quotient:
\[ 9999 \div 360 \approx 27.775 \]
Taking the integer part, we have **27**.

### Step 4: Calculate the greatest four-digit number divisible by 360
Now, we multiply 360 by 27:
\[ 360 \times 27 = 9720 \]

### Step 5: Conclusion
Thus, the greatest four-digit number that is exactly divisible by 15, 24, and 36 is **9720**.

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