If `(8 + 3sqrt(7))^(n) = P + F ` , where P is an integer and F is a proper fraction , then

Options

A. P is a odd integer
B. P is an even integer
C. `F(P + F) = 1`
D. `(1 - F ) (P + F) = 1`

Explanation

To solve the problem, we need to analyze the expression \((8 + 3\sqrt{7})^n\) and express it in the form \(P + F\), where \(P\) is an integer and \(F\) is a proper fraction.

### Step-by-Step Solution:

1. **Understanding the Expression**:
We start with the expression:
\[ (8 + 3\sqrt{7})^n = P + F \]
where \(P\) is an integer and \(F\) is a proper fraction.

2. **Finding the Complement**:
To find \(F\), we consider the conjugate of the expression:
\[ (8 - 3\sqrt{7})^n \]
This will help us find \(F\) because it will yield a similar structure but with a negative term.

3. **Adding the Two Expressions**:
Now we add the two expressions:
\[ (8 + 3\sqrt{7})^n + (8 - 3\sqrt{7})^n = P + F + P' + F' \]
Here, \(P'\) and \(F'\) are the integer and fractional parts of \((8 - 3\sqrt{7})^n\).

4. **Simplifying the Sum**:
The sum simplifies to:
\[ (8 + 3\sqrt{7})^n + (8 - 3\sqrt{7})^n = 2P \]
since the irrational parts (the terms involving \(\sqrt{7}\)) will cancel out.

5. **Identifying the Fractional Parts**:
The fractional parts \(F\) and \(F'\) will also sum up:
\[ F + F' = 1 \]
because both \(F\) and \(F'\) are proper fractions.

6. **Conclusion about \(P\)**:
Since \(2P\) is an integer, \(P\) must also be an integer. However, since \(F + F' = 1\) and both are proper fractions, it follows that:
- \(F\) and \(F'\) are both in the range \(0 < F, F' < 1\).
- Therefore, \(P\) must be odd (since \(P + 1\) is even).

### Final Result:
Thus, we conclude that \(P\) is an odd integer.