When `17^(2010)` is divided by 18, Find the remainder.

Options

A. 1
B. 4
C. 5
D. 3

Answer

To find the remainder when \( 17^{2010} \) is divided by 18, we can use modular arithmetic. Here’s a step-by-step solution:

### Step 1: Understand the problem
We need to calculate \( 17^{2010} \mod 18 \).

### 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 divided by \( 18 \).

### Step 3: Substitute the simplified base into the expression
Now, we can rewrite \( 17^{2010} \) using our simplification:
\[ 17^{2010} \equiv (-1)^{2010} \mod 18 \]

### Step 4: Evaluate the exponent
Since \( 2010 \) is an even number, we know that:
\[ (-1)^{2010} = 1 \]

### Step 5: Conclude the calculation
Thus, we have:
\[ 17^{2010} \equiv 1 \mod 18 \]
This means that when \( 17^{2010} \) is divided by \( 18 \), the remainder is \( 1 \).

### Final Answer
The remainder when \( 17^{2010} \) is divided by \( 18 \) is \( \boxed{1} \).

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