Find the remainder when $(397)^{3589} + 5$ is divided by $398$.

### Concept & Logic This problem is best solved using the Negative Remainder Theorem. When a number $N$ is divided by $N+1$, the remainder can be treated as $-1$. ### Step-by-Step Solution **Given:** Expression: $(397)^{3589} + 5$ Divisor: $398$ **Calculation / Deduction:** * We need to find the remainder of each term when divided by $398$. * First term: $397 \pmod{398}$. Since $397$ is exactly $1$ less than $398$, it leaves a remainder of $-1$. * Raise this negative remainder to the given power: $(-1)^{3589}$. * Since $3589$ is an odd number, $(-1)^{\text{odd}} = -1$. * Second term: $5 \pmod{398}$ is simply $5$, as it is already smaller than the divisor. * Add the remainders together: $-1 + 5 = 4$. ### Exam Strategy & Shortcut Whenever the base is exactly $1$ less than the divisor, look at the power. If the power is odd, the base yields a remainder of $-1$. If the power is even, it yields $+1$. Here, the power is odd, so you get $-1$. Immediately add this to the $+5$ to get the final answer: $4$. This takes less than 5 seconds mentally. ### Common Pitfall A common mistake is trying to apply algebraic expansion formulas like $(x^n + a^n)$ which, while mathematically sound, are overly tedious and increase the chance of making a sign error compared to using direct negative remainders. ### Final Answer Therefore, the correct answer is 4.