`(35)_(n) , (37)_(n), (45)_(n) and (51)_(n)` all are written in base n then the value of n, such that all these four numbers when written in decimal system, then must be the cnsecutive prime numbers:

Options

A. `n = 8`
B. `n = 2`
C. `n = 12`
D. `n = 6`

Answer

To solve the problem, we need to convert each of the numbers given in base \( n \) to decimal and find a value of \( n \) such that the resulting decimal numbers are consecutive prime numbers.

Let's break down the steps:

### Step 1: Convert \( (35)_n \) to Decimal
The number \( (35)_n \) can be converted to decimal as follows:
\[ (35)_n = 3 \cdot n^1 + 5 \cdot n^0 = 3n + 5 \]

### Step 2: Convert \( (37)_n \) to Decimal
Next, we convert \( (37)_n \):
\[ (37)_n = 3 \cdot n^1 + 7 \cdot n^0 = 3n + 7 \]

### Step 3: Convert \( (45)_n \) to Decimal
Now, we convert \( (45)_n \):
\[ (45)_n = 4 \cdot n^1 + 5 \cdot n^0 = 4n + 5 \]

### Step 4: Convert \( (51)_n \) to Decimal
Finally, we convert \( (51)_n \):
\[ (51)_n = 5 \cdot n^1 + 1 \cdot n^0 = 5n + 1 \]

### Step 5: Set Up the Equations
Now we have the following expressions for each number in decimal:
- \( 3n + 5 \)
- \( 3n + 7 \)
- \( 4n + 5 \)
- \( 5n + 1 \)

We need to find \( n \) such that these four expressions yield consecutive prime numbers.

### Step 6: Test Possible Values of \( n \)
We will test the given options for \( n \): 2, 6, 8, and 12.

#### Testing \( n = 2 \):
- \( 3(2) + 5 = 6 + 5 = 11 \)
- \( 3(2) + 7 = 6 + 7 = 13 \)
- \( 4(2) + 5 = 8 + 5 = 13 \)
- \( 5(2) + 1 = 10 + 1 = 11 \) (Not consecutive)

#### Testing \( n = 6 \):
- \( 3(6) + 5 = 18 + 5 = 23 \)
- \( 3(6) + 7 = 18 + 7 = 25 \) (Not prime)

#### Testing \( n = 8 \):
- \( 3(8) + 5 = 24 + 5 = 29 \)
- \( 3(8) + 7 = 24 + 7 = 31 \)
- \( 4(8) + 5 = 32 + 5 = 37 \)
- \( 5(8) + 1 = 40 + 1 = 41 \)

Here, we have the numbers: 29, 31, 37, and 41, which are all consecutive prime numbers.

#### Testing \( n = 12 \):
- \( 3(12) + 5 = 36 + 5 = 41 \)
- \( 3(12) + 7 = 36 + 7 = 43 \)
- \( 4(12) + 5 = 48 + 5 = 53 \)
- \( 5(12) + 1 = 60 + 1 = 61 \)

Here, we have the numbers: 41, 43, 53, and 61, which are also consecutive prime numbers.

### Conclusion
Both \( n = 8 \) and \( n = 12 \) yield consecutive prime numbers, but since the question seems to focus on the first valid option, we conclude that the answer is:

\[ \boxed{8} \]