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

Answer: (B)

$x=(8 \sqrt{3}+13)={ }^{13} C_0 \cdot(8 \sqrt{3})^{13}+{ }^{13} C_1(8 \sqrt{3})^{12}(13)^1+\ldots$
$ x^{\prime}=(8 \sqrt{3}-13)^{13}={ }^{13} C_0(8 \sqrt{3})^{13} - {}^{13} C_1(8 \sqrt{3})^{12}(13)^1+\ldots $
$ x-x^{\prime}=2\left[{ }^{13} C_1 \cdot(8 \sqrt{3})^{12}(13)^1+{ }^{13} C_3(8 \sqrt{3})^{10} \cdot(13)^3 \ldots\right]$
therefore, $x-x^{\prime}$ is even integer, hence $[x]$ is even
$ \text { Now, } y =(7 \sqrt{2}+9)^9={ }^9 C _0(7 \sqrt{2})^9+{ }^9 C _1(7 \sqrt{2})^8(9)^1 +{ }^9 C _2(7 \sqrt{2})^7(9)^2 \ldots \ldots $
$y ^{\prime}=(7 \sqrt{2}-9)^9={ }^9 C _0(7 \sqrt{2})^9-{ }^9 C _1(7 \sqrt{2})^8(9)^1 +{ }^9 C _2(7 \sqrt{2})^7(9)^2 \ldots \ldots $
$ y - y ^{\prime}=2\left[{ }^9 C _1(7 \sqrt{2})^8(9)^1+{ }^9 C _3(7 \sqrt{2})^6(9)^3+\ldots\right] $
$ y - y ^{\prime}=$ Even integer, hence $[ y ] $ is even