If $(8+3\sqrt{7})^n = I + F $ where $ I $ is integer and $F $ is proper fraction, what is $I$? I am beginner to Binomial Theorem and I want to find out weather $I$ is even or odd in $$(8+3\sqrt{7})^n= I+F$$ if it can be expressed as a sum of an Integer $I$ and a proper fraction $F$
How could I find out ?

Notice that 
$$\begin{align}
(8 +3 \sqrt{7})^n + (8 - 3 \sqrt{7})^n &= \sum_{i=0}^n \binom{n}{i} 8 ^i (3 \sqrt{7})^{n-i} + \sum_{i=0}^n \binom{n}{i} (-1)^i 8 ^i (3 \sqrt{7})^{n-i} \\
&= 2 \sum_{i=0}^{\lfloor n/2 \rfloor} \binom{n}{2i} 8^{2i} 3^{2(n-i)} 7^{n-i} \qquad \text{(the odd numbered terms cancel out)}\\
&= 2 N
\end{align}$$
where $N$ is a positive integer.  Since $8 - 3 \sqrt{7} \approx 0.06$, we have
$$(8 +3 \sqrt{7})^n + \epsilon = 2N $$
where $0 < \epsilon < 1$.  
So the integer part of $(8 +3 \sqrt{7})^n$  is odd.
