Find the remainder when $9^6 + 7$ is divided by $8$.

### Concept & Formula Using modular arithmetic, the remainder of a sum is equal to the sum of the individual remainders. Furthermore, any number of the format $(x+1)^n$ divided by $x$ will always leave a remainder of $1^n = 1$. ### Step-by-Step Solution **Given:** * Expression: $9^6 + 7$ * Divisor: $8$ **Calculation / Deduction:** * We need to evaluate the expression modulo $8$: $$ (9^6 + 7) \pmod 8 $$ * Break this into two manageable parts using modular addition properties: $$ (9^6 \pmod 8) + (7 \pmod 8) $$ * **Part 1:** Evaluate $9^6 \pmod 8$. Since $9$ is exactly $1$ more than $8$, the base $9$ leaves a remainder of $1$ when divided by $8$. $$ 1^6 = 1 $$ * **Part 2:** Evaluate $7 \pmod 8$. Since $7$ is strictly less than $8$, its remainder is just itself. $$ 7 \pmod 8 = 7 $$ * Add the two isolated remainders together: $$ 1 + 7 = 8 $$ * Because the calculated sum ($8$) is equal to the divisor ($8$), we must divide it one final time. $8$ divided by $8$ leaves no remainder. $$ 8 \pmod 8 = 0 $$ ### Exam Strategy & Shortcut Use the Remainder Theorem mentally: Treat the divisor $8$ as $x$. The term $9$ becomes $(x+1)$. The expression is $(x+1)^6 + 7$. The $(x+1)$ term divided by $x$ immediately drops to $1$. Add this to $7$ to get $8$. Since $8$ perfectly divides by $8$, the final remainder is trivially $0$. This takes under 5 seconds. ### Common Pitfall Students often forget the final normalization step. When the sum of the remainders equals the divisor (e.g., $1 + 7 = 8$), they might incorrectly choose $8$ as their final answer. A remainder must always be strictly less than the divisor. ### Final Answer Therefore, the correct answer is 0.