If `25^(25)` is divisible by 26, then find the remainder.

Options

A. 21
B. 23
C. 24
D. 25

Answer

To solve the problem of finding the remainder when \( 25^{25} \) is divided by 26, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**: We need to find the remainder of \( 25^{25} \) when divided by 26.

2. **Identify the Base and Divisor**: Here, the base is 25 and the divisor is 26. Notice that 25 is just 1 less than 26.

3. **Calculate the Positive Remainder**: When we divide 25 by 26, the quotient is 0 (since 26 cannot fit into 25) and the remainder is 25. So, we have:
\[ 25 \mod 26 = 25 \]

4. **Calculate the Negative Remainder**: The negative remainder can be found by subtracting the positive remainder from the divisor:
\[ \text{Negative Remainder} = 25 - 26 = -1 \]

5. **Determine the Remainder for the Power**: Since we are interested in \( 25^{25} \), we will use the negative remainder of -1. The exponent is odd (25), so:
\[ (-1)^{25} = -1 \]

6. **Convert Negative Remainder to Positive**: To convert the negative remainder to a positive one, we add the divisor (26):
\[ -1 + 26 = 25 \]

7. **Final Result**: Therefore, the remainder when \( 25^{25} \) is divided by 26 is:
\[ \text{Remainder} = 25 \]

### Conclusion:
The answer is \( \boxed{25} \).