`(49^(13) - 1)` is exactly divisible by

Options

A. 50
B. 51
C. 29
D. 8

Answer

To determine what `(49^(13) - 1)` is exactly divisible by, we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ 49^{13} - 1 \]
We can rewrite this as:
\[ 49^{13} - 1^{13} \]
This is useful because it allows us to apply the difference of powers formula.

### Step 2: Apply the difference of powers formula
The difference of powers formula states that:
\[ A^n - B^n = (A - B)(A^{n-1} + A^{n-2}B + A^{n-3}B^2 + \ldots + B^{n-1}) \]
In our case, \(A = 49\), \(B = 1\), and \(n = 13\) (which is odd). Thus, we can factor our expression:
\[ 49^{13} - 1^{13} = (49 - 1)(49^{12} + 49^{11} \cdot 1 + 49^{10} \cdot 1^2 + \ldots + 1^{12}) \]

### Step 3: Calculate \(49 - 1\)
Now we calculate:
\[ 49 - 1 = 48 \]
So, we have:
\[ 49^{13} - 1 = 48 \cdot (49^{12} + 49^{11} + 49^{10} + \ldots + 1) \]

### Step 4: Determine the divisors of 48
Since \(49^{13} - 1\) is divisible by 48, we need to find the divisors of 48. The prime factorization of 48 is:
\[ 48 = 2^4 \times 3^1 \]
The divisors of 48 are: 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48.

### Step 5: Check the given options
We need to check which of the provided options is a divisor of 48. The options provided are:
1. 50
2. 51
3. 29
4. 8

Among these, only 8 is a divisor of 48.

### Conclusion
Thus, we conclude that:
\[ 49^{13} - 1 \text{ is exactly divisible by } 8. \]