Find the remainder when `(25)^(25)` is divided by 26.

Options

A. 0
B. 1
C. 25
D. 24

Answer

To find the remainder when \( (25)^{25} \) is divided by 26, we can use the property of modular arithmetic.

### Step-by-Step Solution:

1. **Identify the Base and Modulus**:
We need to find \( (25)^{25} \mod 26 \).

2. **Rewrite the Base**:
Notice that \( 25 \) can be expressed in terms of \( 26 \):
\[ 25 \equiv -1 \mod 26 \]
This means that instead of calculating \( (25)^{25} \), we can calculate \( (-1)^{25} \).

3. **Calculate the Power**:
Since \( 25 \) is an odd number, we have:
\[ (-1)^{25} = -1 \]

4. **Find the Remainder**:
Now, we need to convert \( -1 \) into a positive remainder when divided by \( 26 \):
\[ -1 \equiv 25 \mod 26 \]

5. **Conclusion**:
Therefore, the remainder when \( (25)^{25} \) is divided by \( 26 \) is:
\[ \boxed{25} \]