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 will 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).

Now, we can calculate the LCM:

\[ \text{LCM} = 2^3 \times 3^2 \times 5^1 = 8 \times 9 \times 5 \]

Calculating this step-by-step:

1. \(8 \times 9 = 72\)
2. \(72 \times 5 = 360\)

Thus, 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: Determine how many times 360 fits into 9999

To find the largest multiple of 360 that is less than or equal to 9999, we divide:

\[ 9999 \div 360 \approx 27.775 \]

Taking the integer part, we find that 360 fits **27 times** into 9999.

### Step 4: Calculate the greatest multiple of 360

Now, we multiply 360 by 27:

\[ 360 \times 27 = 9720 \]

### Step 5: Conclusion

The greatest four-digit number that is exactly divisible by 15, 24, and 36 is **9720**.

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