Question

Let $x=(8 \sqrt{3}+13)^{13}$ and $y=(7 \sqrt{2}+9)^9$ If $[t]$ denotes the greatest integer $\leq t$, then

• A. $[x]$ is even but $[y]$ is odd
• B. $[x]+[y]$ is even
• C. $[x]$ and $[y]$ are both odd
• D. $[x]$ is odd but $[y]$ is even

Answer 1

The Correct Option is B

Let \( x = \left( 8\sqrt{3}+13 \right)^{13} \) and expand this expression as: \[ x = \left( 8\sqrt{3}-13 \right)^{13} = 13 C_0 \left( 8\sqrt{3} \right)^{13} + 13 C_1 \left( 8\sqrt{3} \right)^{12} (13) + \cdots \] We get the value of \( x - x' \), which results in an even integer. Therefore, \( \left[ x \right] \) is even. Now for \( y = \left( 7\sqrt{2}+9 \right)^9 \), we expand it similarly as: \[ y' = \left( 7\sqrt{2}-9 \right)^9 = 9 C_0 \left( 7\sqrt{2} \right)^9 + 9 C_1 \left( 7\sqrt{2} \right)^8 (9) + \cdots \] We find that \( y - y' \) is also an even integer, and therefore \( \left[ y \right] \) is even. Finally, since both \( [x] \) and \( [y] \) are even, \( [x] + [y] \) is even.