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

Options

A. 17
B. 16
C. 1
D. 2

Answer

To find the remainder when \( 17^{200} \) is divided by 18, we can follow these steps:

### Step 1: Understand the problem
We need to find the remainder of \( 17^{200} \) when divided by 18. This can be expressed mathematically as \( 17^{200} \mod 18 \).

### Step 2: Simplify the base
First, we can simplify \( 17 \) modulo \( 18 \):
\[ 17 \equiv -1 \mod 18 \]
This means that \( 17 \) is equivalent to \( -1 \) when considered in the context of division by \( 18 \).

### Step 3: Substitute the simplified base
Now we can rewrite \( 17^{200} \) using our simplification:
\[ 17^{200} \equiv (-1)^{200} \mod 18 \]

### Step 4: Calculate the exponent
Next, we calculate \( (-1)^{200} \):
\[ (-1)^{200} = 1 \]
This is because any even power of \(-1\) results in \( 1 \).

### Step 5: Find the final remainder
Now we substitute back into our modulo expression:
\[ 17^{200} \equiv 1 \mod 18 \]
Thus, the remainder when \( 17^{200} \) is divided by \( 18 \) is \( 1 \).

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

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