`(3^(2146))/(17)` find R.

Options

A. 7
B. 8
C. 9
D. 16

Answer

To find the remainder \( r \) when \( \frac{3^{2146}}{17} \), we can use modular arithmetic. Here’s a step-by-step solution:

### Step 1: Simplify the exponent using modular arithmetic
We start by finding \( 3^{2146} \mod 17 \). To do this, we can use Fermat's Little Theorem, which states that if \( p \) is a prime number and \( a \) is an integer not divisible by \( p \), then \( a^{p-1} \equiv 1 \mod p \).

Since 17 is prime and 3 is not divisible by 17:
\[ 3^{16} \equiv 1 \mod 17 \]

### Step 2: Reduce the exponent modulo 16
Next, we need to reduce \( 2146 \) modulo \( 16 \):
\[ 2146 \div 16 = 134 \quad \text{(quotient)} \]
\[ 2146 - (134 \times 16) = 2146 - 2144 = 2 \]
Thus, \( 2146 \mod 16 = 2 \).

### Step 3: Calculate \( 3^2 \mod 17 \)
Now, we can simplify our original expression:
\[ 3^{2146} \equiv 3^2 \mod 17 \]
Calculating \( 3^2 \):
\[ 3^2 = 9 \]

### Step 4: Find the remainder
Now we need to find \( 9 \mod 17 \):
\[ 9 \div 17 = 0 \quad \text{(quotient)} \]
\[ 9 - (0 \times 17) = 9 \]
Thus, the remainder \( r \) when \( 3^{2146} \) is divided by \( 17 \) is:
\[ r = 9 \]

### Final Answer
The remainder \( r \) is \( 9 \).

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