`(3^(162))/(162)` find R.

Options

A. 0
B. 1
C. 2
D. 3

Answer

To find the remainder \( R \) when \( \frac{3^{162}}{162} \) is calculated, we can follow these steps:

### Step-by-Step Solution

1. **Understand the Problem**: We need to find the remainder \( R \) when \( 3^{162} \) is divided by \( 162 \).

2. **Factor 162**: The first step is to factor \( 162 \):
\[ 162 = 2 \times 81 = 2 \times 3^4 \]

3. **Use the Chinese Remainder Theorem**: Since \( 162 \) can be expressed as \( 2 \times 3^4 \), we can find the remainders of \( 3^{162} \) modulo \( 2 \) and \( 81 \) separately.

4. **Calculate \( 3^{162} \mod 2 \)**:
\[ 3 \equiv 1 \mod 2 \]
Therefore,
\[ 3^{162} \mod 2 \equiv 1^{162} \equiv 1 \]

5. **Calculate \( 3^{162} \mod 81 \)**:
Since \( 81 = 3^4 \), we can directly see that:
\[ 3^{162} \equiv 0 \mod 81 \]

6. **Combine the Results**: We have:
\[ 3^{162} \equiv 1 \mod 2 \]
\[ 3^{162} \equiv 0 \mod 81 \]

7. **Finding the Common Solution**: We need to find a number \( x \) such that:
\[ x \equiv 0 \mod 81 \]
\[ x \equiv 1 \mod 2 \]

The numbers that satisfy \( x \equiv 0 \mod 81 \) are \( 0, 81, 162, \ldots \). Among these, the only number that is odd (to satisfy \( x \equiv 1 \mod 2 \)) is \( 81 \).

8. **Conclusion**: Therefore, the remainder \( R \) when \( 3^{162} \) is divided by \( 162 \) is:
\[ R = 81 \]