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