Let x be the greatest number such that when 12085, 16914 and 13841 are divided by it, the remainder in each case is same. The sum of digits of x is:

Options

A. a. 14
B. b. 19
C. c. 13
D. d. 16

Answer

To solve the problem step by step, we need to find the greatest number \( x \) such that when 12085, 16914, and 13841 are divided by \( x \), the remainders are the same.

### Step 1: Calculate the differences between the numbers
First, we will find the differences between the given numbers:

1. \( 16914 - 12085 = 4829 \)
2. \( 16914 - 13841 = 3073 \)
3. \( 13841 - 12085 = 1756 \)

### Step 2: Find the GCD of the differences
Next, we need to find the greatest common divisor (GCD) of the differences we calculated:

- The differences are: 4829, 3073, and 1756.

### Step 3: Calculate the GCD of 1756 and 3073
We can use the Euclidean algorithm to find the GCD:

1. \( 3073 \mod 1756 = 3073 - 1756 = 1317 \)
2. \( 1756 \mod 1317 = 1756 - 1317 = 439 \)
3. \( 1317 \mod 439 = 1317 - 3 \times 439 = 0 \)

So, the GCD of 1756 and 3073 is 439.

### Step 4: Calculate the GCD of 439 and 4829
Now we find the GCD of 439 and 4829:

1. \( 4829 \mod 439 = 4829 - 11 \times 439 = 0 \)

Thus, the GCD of 439 and 4829 is also 439.

### Step 5: Conclusion
The greatest number \( x \) that divides 12085, 16914, and 13841 with the same remainder is 439.

### Step 6: Calculate the sum of the digits of \( x \)
Now we need to find the sum of the digits of \( x = 439 \):

- The digits are 4, 3, and 9.
- Sum = \( 4 + 3 + 9 = 16 \).

### Final Answer
The sum of the digits of \( x \) is **16**.

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