If `17^(200)` is divided by 18, the remainder is

Options

A. 0
B. 1
C. 2
D. 16

Answer

To find the remainder when \( 17^{200} \) is divided by 18, we can use the properties of modular arithmetic.

### Step-by-Step Solution:

**Step 1: Rewrite the base in terms of modulo 18.**
\[ 17 \equiv -1 \mod 18 \]
This means that \( 17 \) is congruent to \( -1 \) when divided by \( 18 \).

**Step 2: Substitute the rewritten base into the exponent.**
\[ 17^{200} \equiv (-1)^{200} \mod 18 \]

**Step 3: Calculate the exponent.**
Since \( 200 \) is an even number:
\[ (-1)^{200} = 1 \]

**Step 4: Conclude the calculation.**
Thus, we have:
\[ 17^{200} \equiv 1 \mod 18 \]

**Step 5: Determine the remainder.**
The remainder when \( 17^{200} \) is divided by \( 18 \) is:
\[ \text{Remainder} = 1 \]

### Final Answer:
The remainder when \( 17^{200} \) is divided by \( 18 \) is \( 1 \).

---