Which of the following represents the largest 4 digit number which can be added to 7,249 in order to make the derived number divisible by each of 12, 14, 21, 33 and 54.

Options

A. 9123
B. 9383
C. 8782
D. 8583

Answer

To solve the problem, we need to find the largest 4-digit number that can be added to 7249 such that the resulting sum is divisible by 12, 14, 21, 33, and 54.

### Step 1: Find the Least Common Multiple (LCM)
First, we need to find the least common multiple (LCM) of the numbers 12, 14, 21, 33, and 54.

- **Prime factorization**:
- 12 = 2² × 3
- 14 = 2 × 7
- 21 = 3 × 7
- 33 = 3 × 11
- 54 = 2 × 3³

- **Taking the highest powers of all prime factors**:
- For 2: highest power is 2² (from 12)
- For 3: highest power is 3³ (from 54)
- For 7: highest power is 7¹ (from 14 and 21)
- For 11: highest power is 11¹ (from 33)

Now we can calculate the LCM:
\[ \text{LCM} = 2² × 3³ × 7¹ × 11¹ = 4 × 27 × 7 × 11 \]
Calculating step by step:
- \(4 × 27 = 108\)
- \(108 × 7 = 756\)
- \(756 × 11 = 8316\)

Thus, the LCM of 12, 14, 21, 33, and 54 is **8316**.

### Step 2: Find the Next Multiple of LCM Greater than 7249
Next, we need to find the smallest multiple of 8316 that is greater than 7249.

To find this, we divide 7249 by 8316:
\[ \frac{7249}{8316} \approx 0.872 \]
The next integer is 1, so we multiply:
\[ 1 × 8316 = 8316 \]

### Step 3: Calculate the Required Addition
Now, we need to find the number that needs to be added to 7249 to reach 8316:
\[ 8316 - 7249 = 1067 \]

### Step 4: Check if 1067 is a 4-Digit Number
Since 1067 is a 4-digit number, we have found our answer.

### Conclusion
The largest 4-digit number that can be added to 7249 to make it divisible by 12, 14, 21, 33, and 54 is **1067**.

---