`(25^(26))/(26)` find R.

Answer

To find the remainder of \( \frac{25^{26}}{26} \), we can use Fermat's Little Theorem. Let's go through the solution step by step:

### Step 1: Identify the components
We need to find \( 25^{26} \mod 26 \).

### Step 2: Apply Fermat's Little Theorem
Fermat's Little Theorem states that if \( p \) is a prime number and \( a \) is an integer not divisible by \( p \), then:
\[ a^{p-1} \equiv 1 \mod p \]
In our case, we can use \( p = 13 \) (a prime factor of 26) and \( a = 25 \).

### Step 3: Check if 25 is divisible by 13
Since \( 25 \mod 13 = 12 \) (because \( 25 - 13 = 12 \)), we can use \( 12 \) instead of \( 25 \).

### Step 4: Apply Fermat's Theorem
According to Fermat's theorem:
\[ 12^{12} \equiv 1 \mod 13 \]
Now, we need to find \( 12^{26} \mod 13 \).

### Step 5: Reduce the exponent
Since \( 26 \mod 12 = 2 \) (because \( 26 = 2 \times 12 + 2 \)), we can simplify:
\[ 12^{26} \equiv 12^{2} \mod 13 \]

### Step 6: Calculate \( 12^2 \)
Now calculate \( 12^2 \):
\[ 12^2 = 144 \]

### Step 7: Find \( 144 \mod 13 \)
Now, we need to find \( 144 \mod 13 \):
\[ 144 \div 13 = 11 \quad \text{(remainder 1)} \]
So, \( 144 \mod 13 = 1 \).

### Step 8: Final calculation for \( 25^{26} \mod 26 \)
Now we need to find \( 25^{26} \mod 26 \):
Since \( 25 \equiv -1 \mod 26 \):
\[ (-1)^{26} \equiv 1 \mod 26 \]

### Step 9: Conclusion
Thus, the remainder when \( 25^{26} \) is divided by \( 26 \) is:
\[ \text{R} = 1 \]