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

Explanation

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

### Step 1: Find the LCM of 15, 24, and 36
To find the least common multiple (LCM), we first need the prime factorization of each number:
- **15** = 3 × 5
- **24** = 2^3 × 3
- **36** = 2^2 × 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:
\[ \text{LCM} = 2^3 \times 3^2 \times 5^1 = 8 \times 9 \times 5 \]
Calculating this:
\[ 8 \times 9 = 72 \]
\[ 72 \times 5 = 360 \]
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: Divide 9999 by the LCM (360)
Now, we need to find how many times 360 fits into 9999:
\[ 9999 \div 360 \approx 27.74 \]
Taking the integer part, we find that 360 fits into 9999 a total of **27 times**.

### Step 4: Calculate the largest multiple of 360 within four digits
Now, we multiply 360 by 27:
\[ 360 \times 27 = 9720 \]

### Step 5: Verify if 9720 is divisible by 15, 24, and 36
To confirm, we can check:
- **9720 ÷ 15 = 648** (exactly divisible)
- **9720 ÷ 24 = 405** (exactly divisible)
- **9720 ÷ 36 = 270** (exactly divisible)

Thus, **9720** is indeed divisible by 15, 24, and 36.

### Final Answer
The greatest four-digit number which is exactly divisible by 15, 24, and 36 is **9720**.

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