Find the remainder when $(257^{166} - 243^{166})$ is divided by $500$.

### Concept & Formula This question relies on standard divisibility rules for algebraic identities. Specifically, for an expression in the form $(x^n - a^n)$, it is always perfectly divisible by $(x + a)$ whenever $n$ is an **even** integer. ### Step-by-Step Solution **Given:** Expression: $257^{166} - 243^{166}$ Divisor: $500$ **Calculation / Deduction:** * Identify the format: The expression matches $x^n - a^n$ where $x = 257$, $a = 243$, and $n = 166$. * Check the exponent: The power $166$ is an even number. * Apply the rule: Because the power is even, the expression is perfectly divisible by $(x + a)$. * Calculate $(x + a)$: $$ 257 + 243 = 500 $$ * Since the expression is perfectly divisible by $500$, and our given divisor is also $500$, there is no remainder. ### Exam Strategy & Shortcut Look at the bases and the divisor. $257 + 243 = 500$. This matches the divisor perfectly. Then, look at the sign (minus) and the power (even). By the $(x^n - a^n)$ rule, an even power with a minus sign is divisible by the sum of the bases. Conclude instantly that the remainder is $0$. ### Common Pitfall A frequent error is confusing the parity rules. Students might mix up $x^n - a^n$ and $x^n + a^n$. Remember: 1. $(x^n - a^n)$ is divisible by $(x - a)$ for ALL $n$. 2. $(x^n - a^n)$ is divisible by $(x + a)$ ONLY for EVEN $n$. 3. $(x^n + a^n)$ is divisible by $(x + a)$ ONLY for ODD $n$. ### Final Answer Therefore, the correct answer is 0.