The greatest four digit number which is exactly divisible by 12, 15, 20 and 54 is:

Options

A. 9990
B. 9346
C. 9828
D. 9720

Answer

To find the greatest four-digit number that is exactly divisible by 12, 15, 20, and 54, we can follow these steps:

### Step 1: Find the Least Common Multiple (LCM) of the numbers
We need to find the LCM of 12, 15, 20, and 54. To do this, we first find the prime factorization of each number:

- **12** = 2² × 3¹
- **15** = 3¹ × 5¹
- **20** = 2² × 5¹
- **54** = 2¹ × 3³

Next, we take the highest power of each prime number that appears in these factorizations:

- For **2**, the highest power is 2² (from 12 and 20).
- For **3**, the highest power is 3³ (from 54).
- For **5**, the highest power is 5¹ (from 15 and 20).

Now, we can calculate the LCM:

\[ \text{LCM} = 2^2 \times 3^3 \times 5^1 = 4 \times 27 \times 5 \]

Calculating this step-by-step:

1. \(4 \times 27 = 108\)
2. \(108 \times 5 = 540\)

Thus, the LCM of 12, 15, 20, and 54 is **540**.

### Step 2: Find the greatest four-digit number
The greatest four-digit number is **9999**. To find the greatest number that is divisible by 540, we divide 9999 by 540 and take the integer part of the result:

\[ 9999 \div 540 \approx 18.5148 \]

Taking the integer part, we have **18**.

### Step 3: Multiply to find the greatest number
Now, we multiply 540 by 18:

\[ 540 \times 18 = 9720 \]

### Conclusion
The greatest four-digit number which is exactly divisible by 12, 15, 20, and 54 is **9720**.

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