If $17^{200}$ is divided by $18$, the remainder is

### Concept & Formula This problem is solved using the **Remainder Theorem** and the concept of **Negative Remainders**. When a number $N$ is divided by a divisor $D$, we can express $N$ in terms of $D$. If the number is slightly less than the divisor, using a negative remainder makes power calculations extremely easy. $$ (ax - 1)^n \pmod x = (-1)^n $$ ### Step-by-Step Solution * **Given:** * The expression is $17^{200}$. * The divisor is $18$. * **Calculation:** * We can rewrite the base $17$ in terms of the divisor $18$: $$ 17 = 18 - 1 $$ * Substitute this into the expression: $$ \frac{(18 - 1)^{200}}{18} $$ * When expanding $(18 - 1)^{200}$ using the binomial theorem, every term will contain a multiple of $18$ except for the last term. * The remainder depends entirely on the last term: $$ \text{Remainder} = (-1)^{200} $$ * Since $200$ is an even power, raising a negative number to an even power results in a positive number: $$ (-1)^{200} = 1 $$ ### Exam Strategy & Shortcut Memorize this standard format: **$\frac{x^n}{x+1}$**. If $n$ is even, the remainder is always **$1$**. If $n$ is odd, the remainder is always **$x$**. Here, $x = 17$ and $n = 200$ (even). Therefore, you can instantly conclude the remainder is $1$ without writing anything down. ### Common Pitfall Students often try to find the cyclicity of $17$, which is a massive waste of time for this specific structure. Cyclicity is useful when the base and divisor don't have a difference of $1$, but whenever you see numbers like $17$ and $18$, immediately look for the negative remainder shortcut. ### Final Answer **Therefore, the correct answer is 1.**