Question

The integral part of $(8+3\sqrt{7})^n$ is

• A. an odd integer
• B. an even integer
• C. zero
• D. nothing can be said.

Answer 1

Answer: (B)

Let $(8+3\sqrt{1})^n = p+f$, where $p\,\in \,I$ and $f$ is a proper fraction and let $(8+3\sqrt{1})^n = f'$, a proper fraction $\left[\because 0 < 8-3\sqrt{7} < 1\right]$
Since $\left(8+3\sqrt{7}\right)^{n}+\left(8-3\sqrt{7}\right)^{n} = p+f+f'$ is an even integer
$\therefore p + 1$ is even
$\therefore p$ is an odd integer