$25^{25}$ is divided by $26$, the remainder is

### Concept & Logic This problem is most easily solved using the concept of negative remainders in modular arithmetic. When dividing a number $N$ by a divisor $D$, if $N$ is exactly 1 less than $D$, we can say the remainder is $-1$. $$ (x - 1)^n \pmod x \equiv (-1)^n \pmod x $$ ### Step-by-Step Solution * **Given:** We need to find the remainder of $25^{25}$ divided by $26$. * **Transform using negative remainder:** Express the base $25$ in terms of the divisor $26$. $$25 = 26 - 1$$ Therefore, dividing $25$ by $26$ leaves a remainder of $-1$. * **Apply the exponent:** Substitute the remainder into the original power. $$(-1)^{25}$$ * **Evaluate:** Since the exponent ($25$) is an odd number, a negative base raised to an odd power remains negative. $$(-1)^{25} = -1$$ * **Convert back to a positive remainder:** A negative remainder of $-1$ is equivalent to subtracting $1$ from the divisor. $$26 - 1 = 25$$ ### Exam Strategy & Shortcut Instantly recognize that $25$ is $(26 - 1)$. Any number in the format $(x - 1)^{\text{odd}} \div x$ will always yield a remainder of $(x - 1)$. You can solve this mentally in two seconds: $25$ is $1$ less than $26$, the power is odd, so the remainder is the base itself, $25$. ### Common Pitfall A very common mistake is stopping at the calculation $(-1)^{25} = -1$ and falsely concluding the answer is $1$ because remainders "must be positive." Always add the divisor back to a negative remainder to find the true positive remainder. ### Final Answer Therefore, the correct answer is **25**.