`If (8+3 sqrt7)^n` =P+F, where P is an integer ad F is a proper fraction then

Options

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

Answer

To solve the problem \( (8 + 3\sqrt{7})^n = P + F \), where \( P \) is an integer and \( F \) is a proper fraction, we can follow these steps:

### Step 1: Define 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.

### Step 2: Introduce the Conjugate
To analyze the expression, we introduce the conjugate:
\[ (8 - 3\sqrt{7})^n \]
This will help us find a relationship between \( P \) and \( F \).

### Step 3: Add the Two Expressions
Now, we add both expressions:
\[ (8 + 3\sqrt{7})^n + (8 - 3\sqrt{7})^n \]
This sum will yield an integer because the irrational parts will cancel out.

### Step 4: Set Up the Equation
Let:
\[ S = (8 + 3\sqrt{7})^n + (8 - 3\sqrt{7})^n \]
Since \( S \) is an integer, we can express it as:
\[ S = P + F + (8 - 3\sqrt{7})^n \]

### Step 5: Analyze the Second Term
The term \( (8 - 3\sqrt{7})^n \) is a proper fraction when \( n \) is positive because \( 8 - 3\sqrt{7} \) is less than 1. Therefore, we can denote this term as \( F' \):
\[ F' = (8 - 3\sqrt{7})^n \]

### Step 6: Combine the Results
Now we can write:
\[ P + F + F' = S \]
Since \( F \) and \( F' \) are both proper fractions (between 0 and 1), their sum \( F + F' \) will be less than 2.

### Step 7: Set Up the Final Equation
From the above, we can conclude:
\[ F + F' = S - P \]
Since \( S \) is an integer and \( P \) is also an integer, \( F + F' \) must also be an integer.

### Step 8: Conclusion About \( P \)
Since \( F + F' \) is a proper fraction, we can conclude that \( P \) must be an odd integer. This is because the sum of two proper fractions can only yield an integer when the integer part is odd.

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