Remainder Calculation Using Modular Arithmetic

We need to find the remainder when $25^{25}$ is divided by 26. This can be represented using modular arithmetic as finding the value of $25^{25} \pmod{26}$.

Applying Modular Properties

1. Express the base in terms of the modulus: Notice that 25 is one less than 26. So, we can write:

   $25 \equiv -1 \pmod{26}$

2. Apply the exponent: Using the property that if $a \equiv b \pmod{m}$, then $a^n \equiv b^n \pmod{m}$, we raise both sides to the power of 25:

   $25^{25} \equiv (-1)^{25} \pmod{26}$

3. Simplify the result: Since 25 is an odd exponent, $(-1)^{25}$ equals -1:

   $25^{25} \equiv -1 \pmod{26}$

4. Convert to the standard remainder: A remainder of -1 is equivalent to $26 - 1$ when dividing by 26. Therefore:

   $25^{25} \equiv 25 \pmod{26}$

Thus, the remainder is 25.