Let n be the value of the least integer `(0 gt n gt 5)` so that `3^(2n)+4` is not prime. What is the value of the remainder when `3^(2n)+4` is divided by n?

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the problem, we need to find the least integer \( n \) such that \( 0 < n < 5 \) and \( 3^{2n} + 4 \) is not a prime number. Then, we will determine the remainder when \( 3^{2n} + 4 \) is divided by \( n \).

### Step 1: Check values of \( n \)

We will evaluate \( 3^{2n} + 4 \) for integers \( n = 1, 2, 3, 4 \).

1. **For \( n = 1 \):**
\[ 3^{2 \cdot 1} + 4 = 3^2 + 4 = 9 + 4 = 13 \]
\( 13 \) is a prime number.

2. **For \( n = 2 \):**
\[ 3^{2 \cdot 2} + 4 = 3^4 + 4 = 81 + 4 = 85 \]
\( 85 \) is not a prime number (it can be factored as \( 5 \times 17 \)).

Since \( n = 2 \) is the least integer for which \( 3^{2n} + 4 \) is not prime, we will use this value for further calculations.

### Step 2: Calculate \( 3^{2n} + 4 \) for \( n = 2 \)

We already found:
\[ 3^{2 \cdot 2} + 4 = 85 \]

### Step 3: Find the remainder when \( 85 \) is divided by \( n \)

Now we need to find the remainder when \( 85 \) is divided by \( n = 2 \):
\[ 85 \div 2 = 42 \quad \text{(quotient)} \]
\[ 85 = 2 \times 42 + r \quad \text{(where \( r \) is the remainder)} \]
Calculating \( r \):
\[ r = 85 - 2 \times 42 = 85 - 84 = 1 \]

### Conclusion

The remainder when \( 3^{2n} + 4 \) is divided by \( n \) (where \( n = 2 \)) is:
\[ \boxed{1} \]